This page was generated from unit-3.8-nonlmin/nonlmin.ipynb.

3.8 Nonlinear minimization problems¶

We consider problems of the form

$\text{find } u \in V \text{ s.t. } E(u) \leq E(v) \quad \forall~ v \in V.$

[1]:

from netgen import gui
from ngsolve import *

Scalar minimization problems¶

As a first example we take $V = H^1_0$ and

$E(u) = \int_{\Omega} \vert \nabla u \vert^2 + u^4 - fu ~dx.$

The minimization is equivalent to solving the nonlinear PDE:

$- \Delta u + 4 u^3 = f \text{ in } \Omega$

We solve the PDE with a Newton iteration.

[2]:

from netgen.geom2d import unit_square

mesh = Mesh (unit_square.GenerateMesh(maxh=0.2))
V = H1(mesh, order=4, dirichlet=[1,2,3,4])
u = V.TrialFunction()

To solve the problem we use the Variation integrator. Based on the symbolic description of the energy functional, it is able to

evaluate the energy functional (Energy)

$E(u) \qquad (E:V \to \mathbb{R})$

compute the Gateau derivative for a given $u$ (Apply):

$A(u)(v) = E'(u)(v) \qquad (A(u): V \to \mathbb{R})$

compute the second derivative (AssembleLinearization)

$(\delta A)(w)(u,v) \qquad (\delta A(w): V\times V \to \mathbb{R})$

[3]:

a = BilinearForm (V, symmetric=True)
a += Variation ( (grad(u)*grad(u) + u**4-u) * dx)

Equivalent to:

a += (2 * grad(u) * grad(v) + 4*u*u*u*v - 1 * v)*dx

(which has the same form as the problems in the nonlinear example)

We recall the Newton iteration (cf. unit-3.7 ) we make the loop:

Given an initial guess $u^0$
loop over $i=0,..$ until convergence:
Compute linearization: $A u^i + \delta `A(u^i) :nbsphinx-math:Delta `u^{i} = 0 $:
- $f^i = A u^i$
- $B^i = \delta A(u^i)$
- Solve $B^i \Delta u^i = -f^i$
Update $u^{i+1} = u^i + \Delta u^{i}$
Evaluate stopping criteria
Evaluate $E(u^{i+1})$

As a stopping criteria we take $\langle A u^i,\Delta u^i \rangle = \langle A u^i, A u^i \rangle_{(B^i)^{-1}}< \varepsilon$ .

[4]:

def SolveNonlinearMinProblem(a,gfu,tol=1e-13,maxits=25):
    res = gfu.vec.CreateVector()
    du  = gfu.vec.CreateVector()

    for it in range(maxits):
        print ("Newton iteration {:3}".format(it),end="")
        print ("energy = {:16}".format(a.Energy(gfu.vec)),end="")

        #solve linearized problem:
        a.Apply (gfu.vec, res)
        a.AssembleLinearization (gfu.vec)
        inv = a.mat.Inverse(V.FreeDofs())
        du.data = inv * res

        #update iteration
        gfu.vec.data -= du

        #stopping criteria
        stopcritval = sqrt(abs(InnerProduct(du,res)))
        print ("<A u",it,", A u",it,">_{-1}^0.5 = ", stopcritval)
        if stopcritval < tol:
            break
        Redraw(blocking=True)

[5]:

gfu = GridFunction (V)
gfu.vec[:] = 0
Draw(gfu,mesh,"u")

SolveNonlinearMinProblem(a,gfu)

print ("energy = ", a.Energy(gfu.vec))

Newton iteration   0energy =              0.0<A u 0 , A u 0 >_{-1}^0.5 =  0.13255958653113628
Newton iteration   1energy = -0.008785678644368577<A u 1 , A u 1 >_{-1}^0.5 =  1.1107598523218892e-05
Newton iteration   2energy = -0.008785678706057953<A u 2 , A u 2 >_{-1}^0.5 =  2.8074447148606424e-13
Newton iteration   3energy = -0.008785678706057952<A u 3 , A u 3 >_{-1}^0.5 =  3.8263624548800244e-17
energy =  -0.008785678706057952

Again, a Newton for minimization is shipped with NGSolve:

[6]:

from ngsolve.solvers import *
gfu.vec[:] = 0
NewtonMinimization(a,gfu)
Redraw()

Newton iteration  0
Energy:  0.0
err =  0.13255958653113628
Newton iteration  1
Energy:  -0.008785678644368576
err =  1.1107598523205834e-05
Newton iteration  2
Energy:  -0.008785678706057953
err =  2.807408826572816e-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 =  0.0011225392091069027
Newton iteration   1energy = 8.895994703925567<A u 1 , A u 1 >_{-1}^0.5 =  5.750701824446662
Newton iteration   2energy = 83.84739001095295<A u 2 , A u 2 >_{-1}^0.5 =  13.487648915254145
Newton iteration   3energy = 4.366054527461529<A u 3 , A u 3 >_{-1}^0.5 =  0.5091448618734582
Newton iteration   4energy = -32.60364818572998<A u 4 , A u 4 >_{-1}^0.5 =  12.568357458890837
Newton iteration   5energy =         7529.625<A u 5 , A u 5 >_{-1}^0.5 =  125.37110295796235
Newton iteration   6energy =            384.0<A u 6 , A u 6 >_{-1}^0.5 =  59.96862254571463
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 1
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 2
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 3
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 4
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 5
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 6
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 7
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 8
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 9
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 10
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 11
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 12
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 13
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 14
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 15
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 16
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 17
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 18
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 19
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 20
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 21
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 22
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 23
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 24
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 25
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 26
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 27
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 28
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 29
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 30
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 31
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 32
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 33
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 34
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 35
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 36
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 37
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 38
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 39
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 40
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 41
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 42
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 43
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 44
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 45
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 46
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 47
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 48
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan
loadstep 49
Newton iteration   0energy =              nan<A u 0 , A u 0 >_{-1}^0.5 =  nan
Newton iteration   1energy =              nan<A u 1 , A u 1 >_{-1}^0.5 =  nan
Newton iteration   2energy =              nan<A u 2 , A u 2 >_{-1}^0.5 =  nan
Newton iteration   3energy =              nan<A u 3 , A u 3 >_{-1}^0.5 =  nan
Newton iteration   4energy =              nan<A u 4 , A u 4 >_{-1}^0.5 =  nan
Newton iteration   5energy =              nan<A u 5 , A u 5 >_{-1}^0.5 =  nan
Newton iteration   6energy =              nan<A u 6 , A u 6 >_{-1}^0.5 =  nan
Newton iteration   7energy =              nan<A u 7 , A u 7 >_{-1}^0.5 =  nan
Newton iteration   8energy =              nan<A u 8 , A u 8 >_{-1}^0.5 =  nan
Newton iteration   9energy =              nan<A u 9 , A u 9 >_{-1}^0.5 =  nan
Newton iteration  10energy =              nan<A u 10 , A u 10 >_{-1}^0.5 =  nan
Newton iteration  11energy =              nan<A u 11 , A u 11 >_{-1}^0.5 =  nan
Newton iteration  12energy =              nan<A u 12 , A u 12 >_{-1}^0.5 =  nan
Newton iteration  13energy =              nan<A u 13 , A u 13 >_{-1}^0.5 =  nan
Newton iteration  14energy =              nan<A u 14 , A u 14 >_{-1}^0.5 =  nan
Newton iteration  15energy =              nan<A u 15 , A u 15 >_{-1}^0.5 =  nan
Newton iteration  16energy =              nan<A u 16 , A u 16 >_{-1}^0.5 =  nan
Newton iteration  17energy =              nan<A u 17 , A u 17 >_{-1}^0.5 =  nan
Newton iteration  18energy =              nan<A u 18 , A u 18 >_{-1}^0.5 =  nan
Newton iteration  19energy =              nan<A u 19 , A u 19 >_{-1}^0.5 =  nan
Newton iteration  20energy =              nan<A u 20 , A u 20 >_{-1}^0.5 =  nan
Newton iteration  21energy =              nan<A u 21 , A u 21 >_{-1}^0.5 =  nan
Newton iteration  22energy =              nan<A u 22 , A u 22 >_{-1}^0.5 =  nan
Newton iteration  23energy =              nan<A u 23 , A u 23 >_{-1}^0.5 =  nan
Newton iteration  24energy =              nan<A u 24 , A u 24 >_{-1}^0.5 =  nan

Allen-Cahn equation¶

The Allen-Cahn equations describe the process of phase separation and is the ( $L^2$ ) gradient-flow equation to the energy

$E(v) = \int_{\Omega} \varepsilon \vert \nabla v \vert^2~+~v^2(1-v^2) ~ dx$

i.e. the solution to the Allen-Cahn equation solves

$\partial_t u = \frac{\delta E}{\delta u}$

The quantity $u$ is an indicator for a phase where $-1$ refers to one phase and $1$ to another phase.

The equation has two driving forces:

$u$ is pulled into one of the two minima ( $-1$ and $1$ ) of the nonlinear term $u^2(1-u^2)$ (separation of the phases)
the diffusion term scaled with $\varepsilon$ enforces a smooth transition between the two phases. $\varepsilon$ determines the size of the transition layer

We use the "SymbolicEnergy" feature to formulate the energy minimization problem and combine it with an implicit Euler discretization:

$M u^{n+1} - M u^n = \Delta t \underbrace{\frac{\delta E}{\delta u}}_{=:A(u)} (u^{n+1})$

which we can interpreted as a nonlinear minimization problem again with the energy

$E^{IE}(v) = \int_{\Omega} \frac{\varepsilon}{2} \vert \nabla v \vert^2~+~v^2(1-v^2) + \frac{1}{2\Delta t} \vert v - u^n \vert^2 ~ dx$

To solve the nonlinear equation at every time step we again rely on Newton’s method.

[11]:

from ngsolve import *

from netgen.geom2d import *

periodic = SplineGeometry()
pnts = [ (0,0), (1,0), (1,1), (0,1) ]
pnums = [periodic.AppendPoint(*p) for p in pnts]
lright = periodic.Append ( ["line", pnums[0], pnums[1]],bc="periodic")
btop = periodic.Append ( ["line", pnums[1], pnums[2]], bc="periodic")
periodic.Append ( ["line", pnums[3], pnums[2]], leftdomain=0, rightdomain=1, copy=lright, bc="periodic")
periodic.Append ( ["line", pnums[0], pnums[3]], leftdomain=0, rightdomain=1, copy=btop, bc="periodic")

mesh = Mesh (periodic.GenerateMesh(maxh=0.2))
V = Periodic(H1(mesh, order=4, dirichlet=[]))
u = V.TrialFunction()

eps = 4e-3
dt = 1e-1
gfu = GridFunction(V)
gfuold = GridFunction(V)
a = BilinearForm (V, symmetric=False)
a += Variation( (eps/2*grad(u)*grad(u) + ((1-u*u)*(1-u*u))
                     + 0.5/dt*(u-gfuold)*(u-gfuold)) * dx)

[12]:

from math import pi
gfu = GridFunction(V)
gfu.Set(sin(2*pi*x))
#gfu.Set(sin(1e8*x)) #<- essentially a random function
Draw(gfu,mesh,"u")
SetVisualization (deformation=True)
t = 0

[13]:

for timestep in range(50):
    gfuold.vec.data = gfu.vec
    SolveNonlinearMinProblem(a,gfu)
    Redraw()
    t += dt
    print("t = ", t)

Newton iteration   0energy = 0.41450192074963627<A u 0 , A u 0 >_{-1}^0.5 =  0.2809270463814763
Newton iteration   1energy = 0.37689785802772935<A u 1 , A u 1 >_{-1}^0.5 =  0.01917437047193968
Newton iteration   2energy = 0.376713421040482<A u 2 , A u 2 >_{-1}^0.5 =  9.62893713460141e-05
Newton iteration   3energy = 0.3767134164045806<A u 3 , A u 3 >_{-1}^0.5 =  2.505456297135119e-09
Newton iteration   4energy = 0.3767134164045804<A u 4 , A u 4 >_{-1}^0.5 =  2.879152907139151e-16
t =  0.1
Newton iteration   0energy = 0.3438812865238595<A u 0 , A u 0 >_{-1}^0.5 =  0.2360393474078293
Newton iteration   1energy = 0.31710944329874896<A u 1 , A u 1 >_{-1}^0.5 =  0.013442999343193839
Newton iteration   2energy = 0.3170188793463517<A u 2 , A u 2 >_{-1}^0.5 =  4.660092984275993e-05
Newton iteration   3energy = 0.3170188782605195<A u 3 , A u 3 >_{-1}^0.5 =  5.768048524954674e-10
Newton iteration   4energy = 0.3170188782605194<A u 4 , A u 4 >_{-1}^0.5 =  2.9260894273693223e-16
t =  0.2
Newton iteration   0energy = 0.2954943565138461<A u 0 , A u 0 >_{-1}^0.5 =  0.1834592593102713
Newton iteration   1energy = 0.2791640303389011<A u 1 , A u 1 >_{-1}^0.5 =  0.008014651163108953
Newton iteration   2energy = 0.2791318699362322<A u 2 , A u 2 >_{-1}^0.5 =  1.6250158077594348e-05
Newton iteration   3energy = 0.27913186980419796<A u 3 , A u 3 >_{-1}^0.5 =  6.857119121520948e-11
Newton iteration   4energy = 0.2791318698041979<A u 4 , A u 4 >_{-1}^0.5 =  3.425751677451675e-16
t =  0.30000000000000004
Newton iteration   0energy = 0.2669278950792051<A u 0 , A u 0 >_{-1}^0.5 =  0.13349658930568908
Newton iteration   1energy = 0.25820503551107055<A u 1 , A u 1 >_{-1}^0.5 =  0.004188513106581762
Newton iteration   2energy = 0.2581962576290237<A u 2 , A u 2 >_{-1}^0.5 =  4.366756818946424e-06
Newton iteration   3energy = 0.25819625761948944<A u 3 , A u 3 >_{-1}^0.5 =  4.8506151756504065e-12
Newton iteration   4energy = 0.25819625761948944<A u 4 , A u 4 >_{-1}^0.5 =  3.270692818955673e-16
t =  0.4
Newton iteration   0energy = 0.2520431142368916<A u 0 , A u 0 >_{-1}^0.5 =  0.0923617548386862
Newton iteration   1energy = 0.24783887867786272<A u 1 , A u 1 >_{-1}^0.5 =  0.0019843467589302586
Newton iteration   2energy = 0.2478369092241835<A u 2 , A u 2 >_{-1}^0.5 =  9.685217011836875e-07
Newton iteration   3energy = 0.2478369092237145<A u 3 , A u 3 >_{-1}^0.5 =  2.3475243686450726e-13
Newton iteration   4energy = 0.2478369092237145<A u 4 , A u 4 >_{-1}^0.5 =  4.091478990789537e-16
t =  0.5
Newton iteration   0energy = 0.24499322447001118<A u 0 , A u 0 >_{-1}^0.5 =  0.06164684190451998
Newton iteration   1energy = 0.24311100787346082<A u 1 , A u 1 >_{-1}^0.5 =  0.0008779623506489554
Newton iteration   2energy = 0.2431106224096559<A u 2 , A u 2 >_{-1}^0.5 =  1.882207876093674e-07
Newton iteration   3energy = 0.24311062240963816<A u 3 , A u 3 >_{-1}^0.5 =  8.833158647850137e-15
t =  0.6
Newton iteration   0energy = 0.24187354814793974<A u 0 , A u 0 >_{-1}^0.5 =  0.040166358756514024
Newton iteration   1energy = 0.24107181028211977<A u 1 , A u 1 >_{-1}^0.5 =  0.0003715367480244044
Newton iteration   2energy = 0.24107174125819159<A u 2 , A u 2 >_{-1}^0.5 =  3.366894132543538e-08
Newton iteration   3energy = 0.24107174125819092<A u 3 , A u 3 >_{-1}^0.5 =  5.170667444471021e-16
t =  0.7
Newton iteration   0energy = 0.24055438873764529<A u 0 , A u 0 >_{-1}^0.5 =  0.025782033859246902
Newton iteration   1energy = 0.24022333391554052<A u 1 , A u 1 >_{-1}^0.5 =  0.0001533121425951339
Newton iteration   2energy = 0.24022332216294007<A u 2 , A u 2 >_{-1}^0.5 =  5.795968689030827e-09
Newton iteration   3energy = 0.2402233221629401<A u 3 , A u 3 >_{-1}^0.5 =  4.2728169999287077e-16
t =  0.7999999999999999
Newton iteration   0energy = 0.24001186185815893<A u 0 , A u 0 >_{-1}^0.5 =  0.01642533238340982
Newton iteration   1energy = 0.23987730469401838<A u 1 , A u 1 >_{-1}^0.5 =  6.283809116190392e-05
Newton iteration   2energy = 0.2398773027196846<A u 2 , A u 2 >_{-1}^0.5 =  1.0154930517833773e-09
Newton iteration   3energy = 0.23987730271968474<A u 3 , A u 3 >_{-1}^0.5 =  4.2447785637812194e-16
t =  0.8999999999999999
Newton iteration   0energy = 0.23979158193635353<A u 0 , A u 0 >_{-1}^0.5 =  0.0104644451566102
Newton iteration   1energy = 0.23973691889059892<A u 1 , A u 1 >_{-1}^0.5 =  2.620620881248091e-05
Newton iteration   2energy = 0.23973691854721466<A u 2 , A u 2 >_{-1}^0.5 =  1.99946037686778e-10
Newton iteration   3energy = 0.23973691854721466<A u 3 , A u 3 >_{-1}^0.5 =  4.6622914796398495e-16
t =  0.9999999999999999
Newton iteration   0energy = 0.23970186094053098<A u 0 , A u 0 >_{-1}^0.5 =  0.0067354158269653675
Newton iteration   1energy = 0.2396792028920519<A u 1 , A u 1 >_{-1}^0.5 =  1.1556763084947655e-05
Newton iteration   2energy = 0.23967920282527239<A u 2 , A u 2 >_{-1}^0.5 =  5.0994625523136895e-11
Newton iteration   3energy = 0.23967920282527244<A u 3 , A u 3 >_{-1}^0.5 =  4.753140457014944e-16
t =  1.0999999999999999
Newton iteration   0energy = 0.23966435830447438<A u 0 , A u 0 >_{-1}^0.5 =  0.004452542361404155
Newton iteration   1energy = 0.23965445350959877<A u 1 , A u 1 >_{-1}^0.5 =  5.694142646691109e-06
Newton iteration   2energy = 0.23965445349338702<A u 2 , A u 2 >_{-1}^0.5 =  1.7725000654854798e-11
Newton iteration   3energy = 0.23965445349338704<A u 3 , A u 3 >_{-1}^0.5 =  4.575123380496435e-16
t =  1.2
Newton iteration   0energy = 0.2396476714273789<A u 0 , A u 0 >_{-1}^0.5 =  0.0030979081418181396
Newton iteration   1energy = 0.23964287580252688<A u 1 , A u 1 >_{-1}^0.5 =  3.2840536759848814e-06
Newton iteration   2energy = 0.2396428757971344<A u 2 , A u 2 >_{-1}^0.5 =  7.767180613953351e-12
Newton iteration   3energy = 0.23964287579713447<A u 3 , A u 3 >_{-1}^0.5 =  4.602476027801862e-16
t =  1.3
Newton iteration   0energy = 0.23963934988363728<A u 0 , A u 0 >_{-1}^0.5 =  0.0023262557807642863
Newton iteration   1energy = 0.23963664545354407<A u 1 , A u 1 >_{-1}^0.5 =  2.218176703891121e-06
Newton iteration   2energy = 0.23963664545108387<A u 2 , A u 2 >_{-1}^0.5 =  3.983621199163418e-12
Newton iteration   3energy = 0.23963664545108396<A u 3 , A u 3 >_{-1}^0.5 =  4.726943393711054e-16
t =  1.4000000000000001
Newton iteration   0energy = 0.2396344792626691<A u 0 , A u 0 >_{-1}^0.5 =  0.0019006816819451206
Newton iteration   1energy = 0.23963267365152277<A u 1 , A u 1 >_{-1}^0.5 =  1.687946260856648e-06
Newton iteration   2energy = 0.23963267365009808<A u 2 , A u 2 >_{-1}^0.5 =  2.298225983627877e-12
Newton iteration   3energy = 0.23963267365009816<A u 3 , A u 3 >_{-1}^0.5 =  4.2428964355810436e-16
t =  1.5000000000000002
Newton iteration   0energy = 0.23963111549003335<A u 0 , A u 0 >_{-1}^0.5 =  0.001662928050485115
Newton iteration   1energy = 0.2396297332233654<A u 1 , A u 1 >_{-1}^0.5 =  1.3855651852244353e-06
Newton iteration   2energy = 0.23962973322240547<A u 2 , A u 2 >_{-1}^0.5 =  1.4586276800162247e-12
Newton iteration   3energy = 0.2396297332224055<A u 3 , A u 3 >_{-1}^0.5 =  4.442772535153597e-16
t =  1.6000000000000003
Newton iteration   0energy = 0.23962848049094196<A u 0 , A u 0 >_{-1}^0.5 =  0.0015187632040509363
Newton iteration   1energy = 0.2396273274180991<A u 1 , A u 1 >_{-1}^0.5 =  1.1887244241211346e-06
Newton iteration   2energy = 0.23962732741739268<A u 2 , A u 2 >_{-1}^0.5 =  1.0010671796870583e-12
Newton iteration   3energy = 0.23962732741739273<A u 3 , A u 3 >_{-1}^0.5 =  4.1283112197683045e-16
t =  1.7000000000000004
Newton iteration   0energy = 0.23962625398674184<A u 0 , A u 0 >_{-1}^0.5 =  0.0014193843873505053
Newton iteration   1energy = 0.2396252468233091<A u 1 , A u 1 >_{-1}^0.5 =  1.0456329094500843e-06
Newton iteration   2energy = 0.23962524682276237<A u 2 , A u 2 >_{-1}^0.5 =  7.301168309533506e-13
Newton iteration   3energy = 0.2396252468227624<A u 3 , A u 3 >_{-1}^0.5 =  4.230656435527581e-16
t =  1.8000000000000005
Newton iteration   0energy = 0.23962429644294758<A u 0 , A u 0 >_{-1}^0.5 =  0.0013419591604963788
Newton iteration   1energy = 0.23962339612718064<A u 1 , A u 1 >_{-1}^0.5 =  9.330532164894391e-07
Newton iteration   2energy = 0.23962339612674535<A u 2 , A u 2 >_{-1}^0.5 =  5.563672460999032e-13
Newton iteration   3energy = 0.23962339612674535<A u 3 , A u 3 >_{-1}^0.5 =  4.1802108391138515e-16
t =  1.9000000000000006
Newton iteration   0energy = 0.23962254072403222<A u 0 , A u 0 >_{-1}^0.5 =  0.0012763226584944338
Newton iteration   1energy = 0.2396217263037926<A u 1 , A u 1 >_{-1}^0.5 =  8.399033658728628e-07
Newton iteration   2energy = 0.23962172630343992<A u 2 , A u 2 >_{-1}^0.5 =  4.369413483387965e-13
Newton iteration   3energy = 0.23962172630343986<A u 3 , A u 3 >_{-1}^0.5 =  3.9789823820485357e-16
t =  2.0000000000000004
Newton iteration   0energy = 0.23962094959676444<A u 0 , A u 0 >_{-1}^0.5 =  0.0012179715961910898
Newton iteration   1energy = 0.23962020792824282<A u 1 , A u 1 >_{-1}^0.5 =  7.604668959830782e-07
Newton iteration   2energy = 0.23962020792795358<A u 2 , A u 2 >_{-1}^0.5 =  3.5034459028090097e-13
Newton iteration   3energy = 0.23962020792795358<A u 3 , A u 3 >_{-1}^0.5 =  4.1060598484624654e-16
t =  2.1000000000000005
Newton iteration   0energy = 0.23961949892644918<A u 0 , A u 0 >_{-1}^0.5 =  0.0011648358763700442
Newton iteration   1energy = 0.23961882055016764<A u 1 , A u 1 >_{-1}^0.5 =  6.914942047679861e-07
Newton iteration   2energy = 0.2396188205499285<A u 2 , A u 2 >_{-1}^0.5 =  2.849162416237814e-13
Newton iteration   3energy = 0.2396188205499285<A u 3 , A u 3 >_{-1}^0.5 =  4.081589752428042e-16
t =  2.2000000000000006
Newton iteration   0energy = 0.23961817091741755<A u 0 , A u 0 >_{-1}^0.5 =  0.0011158806899804793
Newton iteration   1energy = 0.23961754835794166<A u 1 , A u 1 >_{-1}^0.5 =  6.309368258835988e-07
Newton iteration   2energy = 0.23961754835774265<A u 2 , A u 2 >_{-1}^0.5 =  2.3424232264681923e-13
Newton iteration   3energy = 0.2396175483577427<A u 3 , A u 3 >_{-1}^0.5 =  4.02045381786918e-16
t =  2.3000000000000007
Newton iteration   0energy = 0.23961695130021168<A u 0 , A u 0 >_{-1}^0.5 =  0.0010705168654729907
Newton iteration   1energy = 0.2396163783254564<A u 1 , A u 1 >_{-1}^0.5 =  5.773776869350729e-07
Newton iteration   2energy = 0.23961637832528954<A u 2 , A u 2 >_{-1}^0.5 =  1.9413331938642087e-13
Newton iteration   3energy = 0.2396163783252895<A u 3 , A u 3 >_{-1}^0.5 =  4.0444178712002976e-16
t =  2.400000000000001
Newton iteration   0energy = 0.23961582808585064<A u 0 , A u 0 >_{-1}^0.5 =  0.0010283540627867244
Newton iteration   1energy = 0.23961529935305312<A u 1 , A u 1 >_{-1}^0.5 =  5.297627567699776e-07
Newton iteration   2energy = 0.23961529935291284<A u 2 , A u 2 >_{-1}^0.5 =  1.619897329526593e-13
Newton iteration   3energy = 0.2396152993529129<A u 3 , A u 3 >_{-1}^0.5 =  3.94945833350714e-16
t =  2.500000000000001
Newton iteration   0energy = 0.23961479095503696<A u 0 , A u 0 >_{-1}^0.5 =  0.0009890970729091943
Newton iteration   1energy = 0.23961430181779741<A u 1 , A u 1 >_{-1}^0.5 =  4.87266603539708e-07
Newton iteration   2energy = 0.23961430181767868<A u 2 , A u 2 >_{-1}^0.5 =  1.3594267931983698e-13
Newton iteration   3energy = 0.23961430181767868<A u 3 , A u 3 >_{-1}^0.5 =  4.452859300184737e-16
t =  2.600000000000001
Newton iteration   0energy = 0.2396138309145446<A u 0 , A u 0 >_{-1}^0.5 =  0.0009525015914669477
Newton iteration   1energy = 0.23961337730105706<A u 1 , A u 1 >_{-1}^0.5 =  4.492198186923643e-07
Newton iteration   2energy = 0.2396133773009562<A u 2 , A u 2 >_{-1}^0.5 =  1.147045763438272e-13
Newton iteration   3energy = 0.23961337730095622<A u 3 , A u 3 >_{-1}^0.5 =  4.0440307212800126e-16
t =  2.700000000000001
Newton iteration   0energy = 0.23961294007376727<A u 0 , A u 0 >_{-1}^0.5 =  0.0009183547757933455
Newton iteration   1energy = 0.23961251839967906<A u 1 , A u 1 >_{-1}^0.5 =  4.1506656583042364e-07
Newton iteration   2energy = 0.23961251839959302<A u 2 , A u 2 >_{-1}^0.5 =  9.723965230427498e-14
t =  2.800000000000001
Newton iteration   0energy = 0.23961211148122993<A u 0 , A u 0 >_{-1}^0.5 =  0.0008864662194064525
Newton iteration   1energy = 0.23961171858168204<A u 1 , A u 1 >_{-1}^0.5 =  3.843377620825692e-07
Newton iteration   2energy = 0.23961171858160824<A u 2 , A u 2 >_{-1}^0.5 =  8.280164617983524e-14
t =  2.9000000000000012
Newton iteration   0energy = 0.23961133899534895<A u 0 , A u 0 >_{-1}^0.5 =  0.0008566633749454628
Newton iteration   1energy = 0.23961097206924217<A u 1 , A u 1 >_{-1}^0.5 =  3.566329547704398e-07
Newton iteration   2energy = 0.2396109720691786<A u 2 , A u 2 >_{-1}^0.5 =  7.079686945960884e-14
t =  3.0000000000000013
Newton iteration   0energy = 0.23961061717765034<A u 0 , A u 0 >_{-1}^0.5 =  0.0008287889263658265
Newton iteration   1energy = 0.23961027374068294<A u 1 , A u 1 >_{-1}^0.5 =  3.316073784608656e-07
Newton iteration   2energy = 0.23961027374062788<A u 2 , A u 2 >_{-1}^0.5 =  6.080527623055362e-14
t =  3.1000000000000014
Newton iteration   0energy = 0.23960994120243923<A u 0 , A u 0 >_{-1}^0.5 =  0.000802699050680157
Newton iteration   1energy = 0.23960961904696845<A u 1 , A u 1 >_{-1}^0.5 =  3.0896230213545007e-07
Newton iteration   2energy = 0.23960961904692074<A u 2 , A u 2 >_{-1}^0.5 =  5.243488745431695e-14
t =  3.2000000000000015
Newton iteration   0energy = 0.23960930677943382<A u 0 , A u 0 >_{-1}^0.5 =  0.0007782621114855675
Newton iteration   1energy = 0.23960900393990814<A u 1 , A u 1 >_{-1}^0.5 =  2.884375804352646e-07
Newton iteration   2energy = 0.2396090039398666<A u 2 , A u 2 >_{-1}^0.5 =  4.541584258742321e-14
t =  3.3000000000000016
Newton iteration   0energy = 0.23960871008705903<A u 0 , A u 0 >_{-1}^0.5 =  0.0007553575814414138
Newton iteration   1energy = 0.23960842481012085<A u 1 , A u 1 >_{-1}^0.5 =  2.6980575461023754e-07
Newton iteration   2energy = 0.23960842481008443<A u 2 , A u 2 >_{-1}^0.5 =  3.9409958947051385e-14
t =  3.4000000000000017
Newton iteration   0energy = 0.23960814771471367<A u 0 , A u 0 >_{-1}^0.5 =  0.0007338751008247049
Newton iteration   1energy = 0.23960787843327264<A u 1 , A u 1 >_{-1}^0.5 =  2.5286728189537163e-07
Newton iteration   2energy = 0.2396078784332407<A u 2 , A u 2 >_{-1}^0.5 =  3.4408330227882534e-14
t =  3.5000000000000018
Newton iteration   0energy = 0.23960761661268298<A u 0 , A u 0 >_{-1}^0.5 =  0.0007137136279155172
Newton iteration   1energy = 0.23960736192339588<A u 1 , A u 1 >_{-1}^0.5 =  2.3744661569880896e-07
Newton iteration   2energy = 0.23960736192336776<A u 2 , A u 2 >_{-1}^0.5 =  3.012187221841644e-14
t =  3.600000000000002
Newton iteration   0energy = 0.23960711404861648<A u 0 , A u 0 >_{-1}^0.5 =  0.0006947806591340681
Newton iteration   1energy = 0.2396068726922975<A u 1 , A u 1 >_{-1}^0.5 =  2.233889422503455e-07
Newton iteration   2energy = 0.2396068726922725<A u 2 , A u 2 >_{-1}^0.5 =  2.6485714180516673e-14
t =  3.700000000000002
Newton iteration   0energy = 0.23960663756965972<A u 0 , A u 0 >_{-1}^0.5 =  0.0006769915073045234
Newton iteration   1energy = 0.23960640841422345<A u 1 , A u 1 >_{-1}^0.5 =  2.105574362009591e-07
Newton iteration   2energy = 0.23960640841420122<A u 2 , A u 2 >_{-1}^0.5 =  2.3373279941882312e-14
t =  3.800000000000002
Newton iteration   0energy = 0.2396061849694745<A u 0 , A u 0 >_{-1}^0.5 =  0.0006602686315034387
Newton iteration   1energy = 0.2396059669950674<A u 1 , A u 1 >_{-1}^0.5 =  1.9883093311422275e-07
Newton iteration   2energy = 0.23960596699504766<A u 2 , A u 2 >_{-1}^0.5 =  2.0633787411416394e-14
t =  3.900000000000002
Newton iteration   0energy = 0.23960575425948266<A u 0 , A u 0 >_{-1}^0.5 =  0.0006445410144735603
Newton iteration   1energy = 0.23960554654551236<A u 1 , A u 1 >_{-1}^0.5 =  1.8810194238216594e-07
Newton iteration   2energy = 0.2396055465454947<A u 2 , A u 2 >_{-1}^0.5 =  1.835368399192164e-14
t =  4.000000000000002
Newton iteration   0energy = 0.23960534364377462<A u 0 , A u 0 >_{-1}^0.5 =  0.0006297435848360659
Newton iteration   1energy = 0.23960514535757993<A u 1 , A u 1 >_{-1}^0.5 =  1.782749416564178e-07
Newton iteration   2energy = 0.23960514535756391<A u 2 , A u 2 >_{-1}^0.5 =  1.6359103102461744e-14
t =  4.100000000000001
Newton iteration   0energy = 0.2396049514971884<A u 0 , A u 0 >_{-1}^0.5 =  0.0006158166819334117
Newton iteration   1energy = 0.23960476188413699<A u 1 , A u 1 >_{-1}^0.5 =  1.6926490630490743e-07
Newton iteration   2energy = 0.23960476188412264<A u 2 , A u 2 >_{-1}^0.5 =  1.4676267063212065e-14
t =  4.200000000000001
Newton iteration   0energy = 0.23960457634614388<A u 0 , A u 0 >_{-1}^0.5 =  0.0006027055613948719
Newton iteration   1energy = 0.23960439472096542<A u 1 , A u 1 >_{-1}^0.5 =  1.609960364893923e-07
Newton iteration   2energy = 0.23960439472095246<A u 2 , A u 2 >_{-1}^0.5 =  1.3180784490767689e-14
t =  4.300000000000001
Newton iteration   0energy = 0.23960421685186517<A u 0 , A u 0 >_{-1}^0.5 =  0.0005903599396007565
Newton iteration   1energy = 0.23960404259105916<A u 1 , A u 1 >_{-1}^0.5 =  1.534006526331149e-07
Newton iteration   2energy = 0.23960404259104737<A u 2 , A u 2 >_{-1}^0.5 =  1.1869329658765281e-14
t =  4.4
Newton iteration   0energy = 0.2396038717956786<A u 0 , A u 0 >_{-1}^0.5 =  0.0005787335752195352
Newton iteration   1energy = 0.2396037043308545<A u 1 , A u 1 >_{-1}^0.5 =  1.4641823370322793e-07
Newton iteration   2energy = 0.23960370433084377<A u 2 , A u 2 >_{-1}^0.5 =  1.0753756111332469e-14
t =  4.5
Newton iteration   0energy = 0.23960354006611326<A u 0 , A u 0 >_{-1}^0.5 =  0.0005677838859563648
Newton iteration   1energy = 0.23960337887814323<A u 1 , A u 1 >_{-1}^0.5 =  1.3999457887475457e-07
Newton iteration   2energy = 0.2396033788781335<A u 2 , A u 2 >_{-1}^0.5 =  9.777453536907239e-15
t =  4.6
Newton iteration   0energy = 0.23960322064757067<A u 0 , A u 0 >_{-1}^0.5 =  0.0005574715986083213
Newton iteration   1energy = 0.23960306526144645<A u 1 , A u 1 >_{-1}^0.5 =  1.340810747023176e-07
Newton iteration   2energy = 0.23960306526143751<A u 2 , A u 2 >_{-1}^0.5 =  8.860077722334228e-15
t =  4.699999999999999
Newton iteration   0energy = 0.2396029126103566<A u 0 , A u 0 >_{-1}^0.5 =  0.0005477604304871145
Newton iteration   1energy = 0.23960276259066193<A u 1 , A u 1 >_{-1}^0.5 =  1.2863405340764783e-07
Newton iteration   2energy = 0.23960276259065366<A u 2 , A u 2 >_{-1}^0.5 =  8.16991512749052e-15
t =  4.799999999999999
Newton iteration   0energy = 0.23960261510190256<A u 0 , A u 0 >_{-1}^0.5 =  0.0005386168002517367
Newton iteration   1energy = 0.2396024700488198<A u 1 , A u 1 >_{-1}^0.5 =  1.236142308543366e-07
Newton iteration   2energy = 0.23960247004881205<A u 2 , A u 2 >_{-1}^0.5 =  7.473122783466527e-15
t =  4.899999999999999
Newton iteration   0energy = 0.23960232733901998<A u 0 , A u 0 >_{-1}^0.5 =  0.0005300095661930661
Newton iteration   1energy = 0.2396021868848036<A u 1 , A u 1 >_{-1}^0.5 =  1.1898621272616119e-07
Newton iteration   2energy = 0.23960218688479662<A u 2 , A u 2 >_{-1}^0.5 =  6.888807762002062e-15
t =  4.999999999999998

Minimal energy extension (postscript in unit-2.1.3 )¶

$u \in V^{ho,disc}, \quad u^{lo,cont} \in V^{lo,cont}, \quad \lambda \in V^{lo,disc},$

[14]:

from netgen.geom2d import unit_square
mesh = Mesh(unit_square.GenerateMesh(maxh=0.1))
fes_ho = Discontinuous(H1(mesh, order=10))
fes_lo = H1(mesh, order=1, dirichlet=".*")
fes_lam = Discontinuous(H1(mesh, order=1))
fes = FESpace([fes_ho, fes_lo, fes_lam])
uho, ulo, lam = fes.TrialFunction()

$\int_{\Omega} \frac12 \Vert \nabla u \Vert^2 - u + \sum_T \sum_{V \in V(T)} ((u-u^{lo})\cdot \lambda)|_{V} \longrightarrow \operatorname{min}!$

[15]:

a = BilinearForm(fes)
a += Variation(0.5 * grad(uho)*grad(uho)*dx
               - 1*uho*dx
               + (uho-ulo)*lam*dx(element_vb=BBND))
gfu = GridFunction(fes)
solvers.Newton(a=a, u=gfu)
Draw(gfu.components[0])

Newton iteration  0
err =  0.3937351156371161
Newton iteration  1
err =  3.874027236775335e-15

The minimization problem is solved by the solution of the PDE:

$\begin{align}\begin{aligned} \int_{\Omega} \nabla u \cdot \nabla v = \int_{\Omega} 1 \cdot v \quad \forall ~ v \in V^{ho,disc}\\under the constraint\end{aligned}\end{align}$

$u(v) = u^{lo}(v) \quad \text{ for all vertices } v \in V(T) \text{ for all } T.$