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3.8 Nonlinear minimization problems

We consider problems of the form

\[\text{find } u \in V \text{ s.t. } E(u) \leq E(v) \quad \forall~ v \in V.\]
In [1]:

import netgen.gui
%gui tk
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.

In [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 SymbolicEnergy 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)
\[(\delta A)(w)(u,v) \qquad (\delta A(w): V\times V \to \mathbb{R})\]
In [3]:

a = BilinearForm (V, symmetric=False)
a += SymbolicEnergy ( grad(u)*grad(u) + u**4-u )

Equivalent to:

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

(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)

    \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 \(\langle A u^i,\Delta u^i \rangle = \langle A u^i, A u^i \rangle_{(B^i)^{-1}}< \varepsilon\).

In [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)

In [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 =  0.1325595815712792
Newton iteration   1energy = -0.008785677986915054<A u 1 , A u 1 >_{-1}^0.5 =  1.1107597333627496e-05
Newton iteration   2energy = -0.008785678048604416<A u 2 , A u 2 >_{-1}^0.5 =  2.807506601278265e-13
Newton iteration   3energy = -0.008785678048604414<A u 3 , A u 3 >_{-1}^0.5 =  3.069514454612314e-17
energy =  -0.008785678048604414

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.

In [6]:

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))


In [7]:

# 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) )

In [8]:

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 + u.Deriv()   # attention: row .. component, col .. derivative
C = F * F.trans

factor = Parameter(1.0)

a = BilinearForm(V, symmetric=False)
a += SymbolicEnergy(  NeoHook (C).Compile() )
a += SymbolicEnergy(  (-factor * InnerProduct(force,u) ).Compile() )

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.

In [9]:

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 = 8.749999999999996<A u 0 , A u 0 >_{-1}^0.5 =  0.016678457857220372
Newton iteration   1energy = 8.750132589925284<A u 1 , A u 1 >_{-1}^0.5 =  0.023333465280071634
Newton iteration   2energy = 8.749861149642193<A u 2 , A u 2 >_{-1}^0.5 =  0.000102324871854834
Newton iteration   3energy = 8.749861144406996<A u 3 , A u 3 >_{-1}^0.5 =  5.1656126548647754e-08
Newton iteration   4energy = 8.749861144407008<A u 4 , A u 4 >_{-1}^0.5 =  2.7534833076896285e-13
Newton iteration   5energy = 8.749861144407008<A u 5 , A u 5 >_{-1}^0.5 =  7.942625526684423e-16
loadstep 1
Newton iteration   0energy = 8.749583890792827<A u 0 , A u 0 >_{-1}^0.5 =  0.016596122310545317
Newton iteration   1energy = 8.749710095753294<A u 1 , A u 1 >_{-1}^0.5 =  0.022958368235712307
Newton iteration   2energy = 8.74944729855201<A u 2 , A u 2 >_{-1}^0.5 =  0.00014216311244477864
Newton iteration   3energy = 8.749447288552698<A u 3 , A u 3 >_{-1}^0.5 =  2.3257814734367656e-06
Newton iteration   4energy = 8.749447288549993<A u 4 , A u 4 >_{-1}^0.5 =  6.330184125593997e-10
Newton iteration   5energy = 8.749447288549998<A u 5 , A u 5 >_{-1}^0.5 =  1.3145920159023592e-15
loadstep 2
Newton iteration   0energy = 8.748898146779494<A u 0 , A u 0 >_{-1}^0.5 =  0.01635691851768748
Newton iteration   1energy = 8.74900526217155<A u 1 , A u 1 >_{-1}^0.5 =  0.02189268120656001
Newton iteration   2energy =  8.7487662601439<A u 2 , A u 2 >_{-1}^0.5 =  0.00020822868315535764
Newton iteration   3energy = 8.748766238787264<A u 3 , A u 3 >_{-1}^0.5 =  4.891163159250807e-06
Newton iteration   4energy = 8.748766238775303<A u 4 , A u 4 >_{-1}^0.5 =  2.6654685723972467e-09
Newton iteration   5energy = 8.748766238775298<A u 5 , A u 5 >_{-1}^0.5 =  1.9181997773232907e-15
loadstep 3
Newton iteration   0energy = 8.747955305248166<A u 0 , A u 0 >_{-1}^0.5 =  0.015981995303982335
Newton iteration   1energy = 8.748035431072346<A u 1 , A u 1 >_{-1}^0.5 =  0.020292141712791756
Newton iteration   2energy = 8.747830050915914<A u 2 , A u 2 >_{-1}^0.5 =  0.00026380941255572184
Newton iteration   3energy = 8.74783001660982<A u 3 , A u 3 >_{-1}^0.5 =  7.247527319852685e-06
Newton iteration   4energy = 8.747830016583555<A u 4 , A u 4 >_{-1}^0.5 =  4.744276205791727e-09
Newton iteration   5energy = 8.747830016583547<A u 5 , A u 5 >_{-1}^0.5 =  3.426563064439674e-15
loadstep 4
Newton iteration   0energy = 8.74677103329084<A u 0 , A u 0 >_{-1}^0.5 =  0.015500978426366845
Newton iteration   1energy = 8.746821875264395<A u 1 , A u 1 >_{-1}^0.5 =  0.018359632535331443
Newton iteration   2energy = 8.74665370335822<A u 2 , A u 2 >_{-1}^0.5 =  0.0002990660120298062
Newton iteration   3energy = 8.746653659164602<A u 3 , A u 3 >_{-1}^0.5 =  8.720462079331242e-06
Newton iteration   4energy = 8.74665365912657<A u 4 , A u 4 >_{-1}^0.5 =  5.242979774834111e-09
Newton iteration   5energy = 8.746653659126574<A u 5 , A u 5 >_{-1}^0.5 =  3.97720967385856e-15
loadstep 5
Newton iteration   0energy = 8.745362746174253<A u 0 , A u 0 >_{-1}^0.5 =  0.014946136334759453
Newton iteration   1energy = 8.745386494953838<A u 1 , A u 1 >_{-1}^0.5 =  0.016292705582918978
Newton iteration   2energy = 8.745254015487305<A u 2 , A u 2 >_{-1}^0.5 =  0.0003132045535723567
Newton iteration   3energy = 8.745253966886729<A u 3 , A u 3 >_{-1}^0.5 =  9.042883675557174e-06
Newton iteration   4energy = 8.745253966845835<A u 4 , A u 4 >_{-1}^0.5 =  4.2168701438486075e-09
Newton iteration   5energy = 8.745253966845846<A u 5 , A u 5 >_{-1}^0.5 =  3.598406255743586e-15
loadstep 6
Newton iteration   0energy = 8.74374841975918<A u 0 , A u 0 >_{-1}^0.5 =  0.014347613422509365
Newton iteration   1energy = 8.743749808685779<A u 1 , A u 1 >_{-1}^0.5 =  0.014248822606138294
Newton iteration   2energy = 8.743648449537007<A u 2 , A u 2 >_{-1}^0.5 =  0.00030934466833337424
Newton iteration   3energy = 8.743648402011962<A u 3 , A u 3 >_{-1}^0.5 =  8.384742090565533e-06
Newton iteration   4energy = 8.743648401976813<A u 4 , A u 4 >_{-1}^0.5 =  2.693807895239245e-09
Newton iteration   5energy = 8.743648401976808<A u 5 , A u 5 >_{-1}^0.5 =  3.709720057706471e-15
loadstep 7
Newton iteration   0energy = 8.74194570825174<A u 0 , A u 0 >_{-1}^0.5 =  0.0137306209358126
Newton iteration   1energy = 8.741930247461754<A u 1 , A u 1 >_{-1}^0.5 =  0.012333315340035126
Newton iteration   2energy = 8.741854283366798<A u 2 , A u 2 >_{-1}^0.5 =  0.0002921290506727813
Newton iteration   3energy = 8.741854240900244<A u 3 , A u 3 >_{-1}^0.5 =  7.129861729080879e-06
Newton iteration   4energy = 8.741854240874824<A u 4 , A u 4 >_{-1}^0.5 =  1.443342960710022e-09
Newton iteration   5energy = 8.74185424087483<A u 5 , A u 5 >_{-1}^0.5 =  4.016975656450898e-15
loadstep 8
Newton iteration   0energy = 8.739971370076443<A u 0 , A u 0 >_{-1}^0.5 =  0.013114473688276701
Newton iteration   1energy = 8.739944202308775<A u 1 , A u 1 >_{-1}^0.5 =  0.010603784256638427
Newton iteration   2energy = 8.739888031385908<A u 2 , A u 2 >_{-1}^0.5 =  0.0002663171865137599
Newton iteration   3energy = 8.739887996038714<A u 3 , A u 3 >_{-1}^0.5 =  5.662593370338205e-06
Newton iteration   4energy = 8.739887996022675<A u 4 , A u 4 >_{-1}^0.5 =  6.719431899287913e-10
Newton iteration   5energy = 8.73988799602267<A u 5 , A u 5 >_{-1}^0.5 =  4.3773856090675734e-15
loadstep 9
Newton iteration   0energy = 8.73784094712138<A u 0 , A u 0 >_{-1}^0.5 =  0.01251287296514784
Newton iteration   1energy = 8.737806316551259<A u 1 , A u 1 >_{-1}^0.5 =  0.009081886577209623
Newton iteration   2energy = 8.737765099398299<A u 2 , A u 2 >_{-1}^0.5 =  0.00023603860491985037
Newton iteration   3energy = 8.737765071600855<A u 3 , A u 3 >_{-1}^0.5 =  4.258372559407323e-06
Newton iteration   4energy = 8.73776507159178<A u 4 , A u 4 >_{-1}^0.5 =  2.778505250314751e-10
Newton iteration   5energy = 8.73776507159178<A u 5 , A u 5 >_{-1}^0.5 =  6.056468230296881e-15
loadstep 10
Newton iteration   0energy = 8.735568627127268<A u 0 , A u 0 >_{-1}^0.5 =  0.011934806211101961
Newton iteration   1energy = 8.735529764422171<A u 1 , A u 1 >_{-1}^0.5 =  0.0077659135356494555
Newton iteration   2energy = 8.73549961771559<A u 2 , A u 2 >_{-1}^0.5 =  0.00020449613889987533
Newton iteration   3energy = 8.735499596834673<A u 3 , A u 3 >_{-1}^0.5 =  3.0647929288143972e-06
Newton iteration   4energy = 8.735499596829976<A u 4 , A u 4 >_{-1}^0.5 =  1.0315667610823363e-10
Newton iteration   5energy = 8.73549959682998<A u 5 , A u 5 >_{-1}^0.5 =  4.943459155768916e-15
loadstep 11
Newton iteration   0energy = 8.733167226220827<A u 0 , A u 0 >_{-1}^0.5 =  0.011385614969521708
Newton iteration   1energy = 8.73312644913484<A u 1 , A u 1 >_{-1}^0.5 =  0.006641005193965611
Newton iteration   2energy = 8.733104397426635<A u 2 , A u 2 >_{-1}^0.5 =  0.00017393963828985667
Newton iteration   3energy = 8.73310438231158<A u 3 , A u 3 >_{-1}^0.5 =  2.1292807624511996e-06
Newton iteration   4energy = 8.733104382309314<A u 4 , A u 4 >_{-1}^0.5 =  3.432001970407436e-11
Newton iteration   5energy = 8.733104382309314<A u 5 , A u 5 >_{-1}^0.5 =  6.309315793552211e-15
loadstep 12
Newton iteration   0energy = 8.730648243500866<A u 0 , A u 0 >_{-1}^0.5 =  0.010867985330305738
Newton iteration   1energy = 8.730607135592724<A u 1 , A u 1 >_{-1}^0.5 =  0.005686263271293424
Newton iteration   2energy = 8.73059096457382<A u 2 , A u 2 >_{-1}^0.5 =  0.00014577993437518897
Newton iteration   3energy = 8.73059095395286<A u 3 , A u 3 >_{-1}^0.5 =  1.4383390443423343e-06
Newton iteration   4energy = 8.73059095395182<A u 4 , A u 4 >_{-1}^0.5 =  9.985478077309379e-12
Newton iteration   5energy = 8.730590953951822<A u 5 , A u 5 >_{-1}^0.5 =  6.304937632385649e-15
loadstep 13
Newton iteration   0energy = 8.728021954318086<A u 0 , A u 0 >_{-1}^0.5 =  0.01038276202251739
Newton iteration   1energy = 8.727981547860873<A u 1 , A u 1 >_{-1}^0.5 =  0.004879172690876309
Newton iteration   2energy = 8.727969638945583<A u 2 , A u 2 >_{-1}^0.5 =  0.00012076034968403019
Newton iteration   3energy = 8.727969631655753<A u 3 , A u 3 >_{-1}^0.5 =  9.504870926466129e-07
Newton iteration   4energy = 8.727969631655302<A u 4 , A u 4 >_{-1}^0.5 =  2.3259742706803135e-12
Newton iteration   5energy = 8.727969631655306<A u 5 , A u 5 >_{-1}^0.5 =  7.021360363946004e-15
loadstep 14
Newton iteration   0energy = 8.72529752061992<A u 0 , A u 0 >_{-1}^0.5 =  0.00992957077965983
Newton iteration   1energy = 8.725258452639936<A u 1 , A u 1 >_{-1}^0.5 =  0.004198069592467644
Newton iteration   2energy = 8.725249634804557<A u 2 , A u 2 >_{-1}^0.5 =  9.913700636600242e-05
Newton iteration   3energy = 8.725249629890909<A u 3 , A u 3 >_{-1}^0.5 =  6.177005897210002e-07
Newton iteration   4energy = 8.725249629890715<A u 4 , A u 4 >_{-1}^0.5 =  2.7075801825504915e-13
Newton iteration   5energy = 8.72524962989072<A u 5 , A u 5 >_{-1}^0.5 =  6.4867867334457204e-15
loadstep 15
Newton iteration   0energy = 8.722483105125077<A u 0 , A u 0 >_{-1}^0.5 =  0.009507271533554363
Newton iteration   1energy = 8.722445738804593<A u 1 , A u 1 >_{-1}^0.5 =  0.003623338365125314
Newton iteration   2energy = 8.722439169043259<A u 2 , A u 2 >_{-1}^0.5 =  8.084105831464236e-05
Newton iteration   3energy = 8.722439165775617<A u 3 , A u 3 >_{-1}^0.5 =  3.9657874014284555e-07
Newton iteration   4energy = 8.72243916577554<A u 4 , A u 4 >_{-1}^0.5 =  1.5322265915943703e-13
Newton iteration   5energy = 8.72243916577554<A u 5 , A u 5 >_{-1}^0.5 =  7.020924862732914e-15
loadstep 16
Newton iteration   0energy = 8.719585981725038<A u 0 , A u 0 >_{-1}^0.5 =  0.009114276766163665
Newton iteration   1energy = 8.719550495831808<A u 1 , A u 1 >_{-1}^0.5 =  0.003137848417803922
Newton iteration   2energy = 8.719545568051617<A u 2 , A u 2 >_{-1}^0.5 =  6.561021540824207e-05
Newton iteration   3energy = 8.719545565899164<A u 3 , A u 3 >_{-1}^0.5 =  2.525155744874241e-07
Newton iteration   4energy = 8.719545565899123<A u 4 , A u 4 >_{-1}^0.5 =  1.4980696449755864e-13
Newton iteration   5energy = 8.719545565899129<A u 5 , A u 5 >_{-1}^0.5 =  7.025195703399348e-15
loadstep 17
Newton iteration   0energy = 8.716612638121687<A u 0 , A u 0 >_{-1}^0.5 =  0.008748768597255036
Newton iteration   1energy = 8.716579090586363<A u 1 , A u 1 >_{-1}^0.5 =  0.0027269718667353913
Newton iteration   2energy = 8.716575368455187<A u 2 , A u 2 >_{-1}^0.5 =  5.30864063971943e-05
Newton iteration   3energy = 8.716575367045994<A u 3 , A u 3 >_{-1}^0.5 =  1.5998581937951101e-07
Newton iteration   4energy = 8.716575367045984<A u 4 , A u 4 >_{-1}^0.5 =  9.24264467888606e-14
loadstep 18
Newton iteration   0energy = 8.713568868939605<A u 0 , A u 0 >_{-1}^0.5 =  0.008408842703937033
Newton iteration   1energy = 8.713537240942847<A u 1 , A u 1 >_{-1}^0.5 =  0.0023783933491091343
Newton iteration   2energy =  8.7135344093602<A u 2 , A u 2 >_{-1}^0.5 =  4.288234198914801e-05
Newton iteration   3energy = 8.713534408440674<A u 3 , A u 3 >_{-1}^0.5 =  1.0113452182494698e-07
Newton iteration   4energy = 8.713534408440665<A u 4 , A u 4 >_{-1}^0.5 =  4.866020392496476e-14
loadstep 19
Newton iteration   0energy = 8.71045985887359<A u 0 , A u 0 >_{-1}^0.5 =  0.008092600787386375
Newton iteration   1energy = 8.710430084840077<A u 1 , A u 1 >_{-1}^0.5 =  0.0020818348293488013
Newton iteration   2energy = 8.710427915258844<A u 2 , A u 2 >_{-1}^0.5 =  3.462274032326428e-05
Newton iteration   3energy = 8.710427914659423<A u 3 , A u 3 >_{-1}^0.5 =  6.393209442938853e-08
Newton iteration   4energy = 8.710427914659427<A u 4 , A u 4 >_{-1}^0.5 =  2.4472323028745738e-14
loadstep 20
Newton iteration   0energy = 8.707290256175286<A u 0 , A u 0 >_{-1}^0.5 =  0.007798207601916305
Newton iteration   1energy = 8.707262243846786<A u 1 , A u 1 >_{-1}^0.5 =  0.0018287624748981241
Newton iteration   2energy = 8.70726056963296<A u 2 , A u 2 >_{-1}^0.5 =  2.7966782719369756e-05
Newton iteration   3energy = 8.707260569241857<A u 3 , A u 3 >_{-1}^0.5 =  4.048825692057199e-08
Newton iteration   4energy = 8.707260569241852<A u 4 , A u 4 >_{-1}^0.5 =  1.2768784682907216e-14
loadstep 21
Newton iteration   0energy = 8.704064237168412<A u 0 , A u 0 >_{-1}^0.5 =  0.007523924008749803
Newton iteration   1energy = 8.704037880818536<A u 1 , A u 1 >_{-1}^0.5 =  0.0016121092918853037
Newton iteration   2energy = 8.704036579777165<A u 2 , A u 2 >_{-1}^0.5 =  2.2617802715322334e-05
Newton iteration   3energy = 8.70403657952137<A u 3 , A u 3 >_{-1}^0.5 =  2.5725058090116162e-08
Newton iteration   4energy = 8.704036579521365<A u 4 , A u 4 >_{-1}^0.5 =  8.839028230895453e-15
loadstep 22
Newton iteration   0energy = 8.700785562650578<A u 0 , A u 0 >_{-1}^0.5 =  0.0072681240911013885
Newton iteration   1energy = 8.700760751620374<A u 1 , A u 1 >_{-1}^0.5 =  0.0014260281489234519
Newton iteration   2energy = 8.700759733595584<A u 2 , A u 2 >_{-1}^0.5 =  1.8325012042621618e-05
Newton iteration   3energy = 8.700759733427669<A u 3 , A u 3 >_{-1}^0.5 =  1.6416821349247635e-08
Newton iteration   4energy = 8.70075973342767<A u 4 , A u 4 >_{-1}^0.5 =  7.0644037247864004e-15
loadstep 23
Newton iteration   0energy = 8.697457627081807<A u 0 , A u 0 >_{-1}^0.5 =  0.007029301882349006
Newton iteration   1energy = 8.69743425115637<A u 1 , A u 1 >_{-1}^0.5 =  0.0012656794747770324
Newton iteration   2energy = 8.69743344921061<A u 2 , A u 2 >_{-1}^0.5 =  1.4880750185375254e-05
Newton iteration   3energy = 8.697433449099881<A u 3 , A u 3 >_{-1}^0.5 =  1.0531732189483211e-08
Newton iteration   4energy = 8.697433449099886<A u 4 , A u 4 >_{-1}^0.5 =  7.324873708442945e-15
loadstep 24
Newton iteration   0energy = 8.694083501431793<A u 0 , A u 0 >_{-1}^0.5 =  0.006806071495723261
Newton iteration   1energy = 8.694061454104311<A u 1 , A u 1 >_{-1}^0.5 =  0.0011270526725460628
Newton iteration   2energy = 8.694060818220638<A u 2 , A u 2 >_{-1}^0.5 =  1.2115589040157976e-05
Newton iteration   3energy = 8.69406081814725<A u 3 , A u 3 >_{-1}^0.5 =  6.7961929805900055e-09
Newton iteration   4energy = 8.694060818147245<A u 4 , A u 4 >_{-1}^0.5 =  7.34815895816638e-15
loadstep 25
Newton iteration   0energy = 8.69066597049359<A u 0 , A u 0 >_{-1}^0.5 =  0.006597163213874111
Newton iteration   1energy = 8.690645150830838<A u 1 , A u 1 >_{-1}^0.5 =  0.0010068179295690766
Newton iteration   2energy = 8.690644643396258<A u 2 , A u 2 >_{-1}^0.5 =  9.892734819757748e-06
Newton iteration   3energy = 8.690644643347325<A u 3 , A u 3 >_{-1}^0.5 =  4.413530878878279e-09
Newton iteration   4energy = 8.690644643347321<A u 4 , A u 4 >_{-1}^0.5 =  8.077944067857594e-15
loadstep 26
Newton iteration   0energy = 8.687207565391779<A u 0 , A u 0 >_{-1}^0.5 =  0.006401417245850291
Newton iteration   1energy = 8.687187878984766<A u 1 , A u 1 >_{-1}^0.5 =  0.0009022042935673603
Newton iteration   2energy = 8.687187471534324<A u 2 , A u 2 >_{-1}^0.5 =  8.102543734498749e-06
Newton iteration   3energy = 8.687187471501492<A u 3 , A u 3 >_{-1}^0.5 =  2.8853474185662717e-09
Newton iteration   4energy = 8.687187471501497<A u 4 , A u 4 >_{-1}^0.5 =  7.88933517416324e-15
loadstep 27
Newton iteration   0energy = 8.683710591929625<A u 0 , A u 0 >_{-1}^0.5 =  0.006217776276937307
Newton iteration   1energy = 8.683691951255541<A u 1 , A u 1 >_{-1}^0.5 =  0.000810899873211364
Newton iteration   2energy =  8.6836916221124<A u 2 , A u 2 >_{-1}^0.5 =  6.6575598591021325e-06
Newton iteration   3energy = 8.68369162209024<A u 3 , A u 3 >_{-1}^0.5 =  1.899274794240032e-09
Newton iteration   4energy = 8.683691622090233<A u 4 , A u 4 >_{-1}^0.5 =  7.480573029729903e-15
loadstep 28
Newton iteration   0energy = 8.680177155339257<A u 0 , A u 0 >_{-1}^0.5 =  0.006045277540623081
Newton iteration   1energy = 8.680159479753097<A u 1 , A u 1 >_{-1}^0.5 =  0.0007309703769520457
Newton iteration   2energy = 8.680159212308286<A u 2 , A u 2 >_{-1}^0.5 =  5.488234785038055e-06
Newton iteration   3energy = 8.68015921229323<A u 3 , A u 3 >_{-1}^0.5 =  1.2589353554913938e-09
Newton iteration   4energy = 8.680159212293228<A u 4 , A u 4 >_{-1}^0.5 =  7.266489767493362e-15
loadstep 29
Newton iteration   0energy = 8.676609181924405<A u 0 , A u 0 >_{-1}^0.5 =  0.005883044873737748
Newton iteration   1energy = 8.67659239742593<A u 1 , A u 1 >_{-1}^0.5 =  0.0006607926970899941
Newton iteration   2energy = 8.676592178876996<A u 2 , A u 2 >_{-1}^0.5 =  4.539348109422138e-06
Newton iteration   3energy = 8.676592178866697<A u 3 , A u 3 >_{-1}^0.5 =  8.403501989065408e-10
Newton iteration   4energy =  8.6765921788667<A u 4 , A u 4 >_{-1}^0.5 =  7.67629239237462e-15
loadstep 30
Newton iteration   0energy = 8.673008438018613<A u 0 , A u 0 >_{-1}^0.5 =  0.005730281036566557
Newton iteration   1energy = 8.672992476890737<A u 1 , A u 1 >_{-1}^0.5 =  0.0005990007596523957
Newton iteration   2energy = 8.672992297311481<A u 2 , A u 2 >_{-1}^0.5 =  3.7670762457130904e-06
Newton iteration   3energy = 8.672992297304384<A u 3 , A u 3 >_{-1}^0.5 =  5.648740373123977e-10
Newton iteration   4energy = 8.672992297304384<A u 4 , A u 4 >_{-1}^0.5 =  8.035640983518844e-15
loadstep 31
Newton iteration   0energy = 8.669376546622743<A u 0 , A u 0 >_{-1}^0.5 =  0.005586260459714786
Newton iteration   1energy = 8.669361347004394<A u 1 , A u 1 >_{-1}^0.5 =  0.0005444413363355358
Newton iteration   2energy = 8.669361198654682<A u 2 , A u 2 >_{-1}^0.5 =  3.136626679136021e-06
Newton iteration   3energy = 8.669361198649762<A u 3 , A u 3 >_{-1}^0.5 =  3.8234116538423575e-10
Newton iteration   4energy = 8.66936119864977<A u 4 , A u 4 >_{-1}^0.5 =  8.562868014308272e-15
loadstep 32
Newton iteration   0energy = 8.665715002034561<A u 0 , A u 0 >_{-1}^0.5 =  0.005450322501437702
Newton iteration   1energy = 8.665700507468294<A u 1 , A u 1 >_{-1}^0.5 =  0.0004961379326198621
Newton iteration   2energy = 8.665700384279145<A u 2 , A u 2 >_{-1}^0.5 =  2.6203476609643125e-06
Newton iteration   3energy = 8.66570038427571<A u 3 , A u 3 >_{-1}^0.5 =  2.6056777347312574e-10
Newton iteration   4energy = 8.665700384275706<A u 4 , A u 4 >_{-1}^0.5 =  9.019629514694351e-15
loadstep 33
Newton iteration   0energy = 8.66202518273859<A u 0 , A u 0 >_{-1}^0.5 =  0.005321865247746466
Newton iteration   1energy = 8.662011341718637<A u 1 , A u 1 >_{-1}^0.5 =  0.0004532612188666138
Newton iteration   2energy = 8.662011238905643<A u 2 , A u 2 >_{-1}^0.5 =  2.1962280959714483e-06
Newton iteration   3energy = 8.662011238903226<A u 3 , A u 3 >_{-1}^0.5 =  1.7877630682069847e-10
Newton iteration   4energy = 8.662011238903228<A u 4 , A u 4 >_{-1}^0.5 =  7.564973133263078e-15
loadstep 34
Newton iteration   0energy = 8.658308362786434<A u 0 , A u 0 >_{-1}^0.5 =  0.005200339855866979
Newton iteration   1energy = 8.658295128322484<A u 1 , A u 1 >_{-1}^0.5 =  0.000415104764483889
Newton iteration   2energy = 8.658295042094178<A u 2 , A u 2 >_{-1}^0.5 =  1.8467124404900584e-06
Newton iteration   3energy = 8.658295042092467<A u 3 , A u 3 >_{-1}^0.5 =  1.234695359286693e-10
Newton iteration   4energy = 8.658295042092472<A u 4 , A u 4 >_{-1}^0.5 =  8.782606917524184e-15
loadstep 35
Newton iteration   0energy = 8.654565721865083<A u 0 , A u 0 >_{-1}^0.5 =  0.005085245422016683
Newton iteration   1energy = 8.654553051070815<A u 1 , A u 1 >_{-1}^0.5 =  0.000381065074801646
Newton iteration   2energy = 8.65455297840727<A u 2 , A u 2 >_{-1}^0.5 =  1.557767056817645e-06
Newton iteration   3energy = 8.654552978406052<A u 3 , A u 3 >_{-1}^0.5 =  8.582529018395146e-11
Newton iteration   4energy = 8.65455297840605<A u 4 , A u 4 >_{-1}^0.5 =  8.081681666189039e-15
loadstep 36
Newton iteration   0energy = 8.650798354223038<A u 0 , A u 0 >_{-1}^0.5 =  0.004976124343087951
Newton iteration   1energy = 8.650786207933697<A u 1 , A u 1 >_{-1}^0.5 =  0.0003506251242975913
Newton iteration   2energy = 8.650786146417616<A u 2 , A u 2 >_{-1}^0.5 =  1.3181457148090865e-06
Newton iteration   3energy = 8.650786146416742<A u 3 , A u 3 >_{-1}^0.5 =  6.003412312235294e-11
Newton iteration   4energy = 8.65078614641675<A u 4 , A u 4 >_{-1}^0.5 =  8.696956839627608e-15
loadstep 37
Newton iteration   0energy = 8.64700727659986<A u 0 , A u 0 >_{-1}^0.5 =  0.004872558136248435
Newton iteration   1energy = 8.646995619020746<A u 1 , A u 1 >_{-1}^0.5 =  0.00032334073662541326
Newton iteration   2energy = 8.646995566707893<A u 2 , A u 2 >_{-1}^0.5 =  1.1188120075819826e-06
Newton iteration   3energy = 8.646995566707279<A u 3 , A u 3 >_{-1}^0.5 =  4.2253743915995646e-11
Newton iteration   4energy = 8.646995566707277<A u 4 , A u 4 >_{-1}^0.5 =  9.377941245241211e-15
loadstep 38
Newton iteration   0energy = 8.64319343528497<A u 0 , A u 0 >_{-1}^0.5 =  0.0047741636782933534
Newton iteration   1energy = 8.643182233670666<A u 1 , A u 1 >_{-1}^0.5 =  0.00029882928765195875
Newton iteration   2energy = 8.643182188990053<A u 2 , A u 2 >_{-1}^0.5 =  9.524850091632305e-07
Newton iteration   3energy = 8.643182188989602<A u 3 , A u 3 >_{-1}^0.5 =  2.991908303311475e-11
Newton iteration   4energy = 8.643182188989597<A u 4 , A u 4 >_{-1}^0.5 =  8.794424361705271e-15
loadstep 39
Newton iteration   0energy = 8.639357712413794<A u 0 , A u 0 >_{-1}^0.5 =  0.004680589826624847
Newton iteration   1energy = 8.639346936777114<A u 1 , A u 1 >_{-1}^0.5 =  0.0002767603087020712
Newton iteration   2energy = 8.639346898453521<A u 2 , A u 2 >_{-1}^0.5 =  8.132815134318612e-07
Newton iteration   3energy = 8.639346898453198<A u 3 , A u 3 >_{-1}^0.5 =  2.1307647885300333e-11
Newton iteration   4energy = 8.639346898453194<A u 4 , A u 4 >_{-1}^0.5 =  8.826338258215783e-15
loadstep 40
Newton iteration   0energy = 8.63550093159491<A u 0 , A u 0 >_{-1}^0.5 =  0.004591514385093225
Newton iteration   1energy = 8.635490554443683<A u 1 , A u 1 >_{-1}^0.5 =  0.0002568476480043424
Newton iteration   2energy = 8.635490521437436<A u 2 , A u 2 >_{-1}^0.5 =  6.96433934687339e-07
Newton iteration   3energy =  8.6354905214372<A u 3 , A u 3 >_{-1}^0.5 =  1.5261591030717674e-11
Newton iteration   4energy = 8.63549052143719<A u 4 , A u 4 >_{-1}^0.5 =  9.275349131281183e-15
loadstep 41
Newton iteration   0energy = 8.631623862948826<A u 0 , A u 0 >_{-1}^0.5 =  0.004506641380400642
Newton iteration   1energy = 8.631613859048633<A u 1 , A u 1 >_{-1}^0.5 =  0.00023884291352208444
Newton iteration   2energy = 8.631613830508467<A u 2 , A u 2 >_{-1}^0.5 =  5.98067464130121e-07
Newton iteration   3energy = 8.63161383050828<A u 3 , A u 3 >_{-1}^0.5 =  1.0991512102948347e-11
Newton iteration   4energy = 8.631613830508293<A u 4 , A u 4 >_{-1}^0.5 =  9.388595441023754e-15
loadstep 42
Newton iteration   0energy = 8.627727227628869<A u 0 , A u 0 >_{-1}^0.5 =  0.0044256986169769995
Newton iteration   1energy = 8.627717573789365<A u 1 , A u 1 >_{-1}^0.5 =  0.0002225299720352437
Newton iteration   2energy = 8.627717549015356<A u 2 , A u 2 >_{-1}^0.5 =  5.150236660184008e-07
Newton iteration   3energy = 8.62771754901523<A u 3 , A u 3 >_{-1}^0.5 =  7.961224220497201e-12
Newton iteration   4energy = 8.62771754901523<A u 4 , A u 4 >_{-1}^0.5 =  9.341742069744362e-15
loadstep 43
Newton iteration   0energy = 8.623811701884499<A u 0 , A u 0 >_{-1}^0.5 =  0.004348435481444442
Newton iteration   1energy = 8.62380237676742<A u 1 , A u 1 >_{-1}^0.5 =  0.00020772032165363919
Newton iteration   2energy = 8.623802355181764<A u 2 , A u 2 >_{-1}^0.5 =  4.44720525750143e-07
Newton iteration   3energy = 8.623802355181667<A u 3 , A u 3 >_{-1}^0.5 =  5.794433094929551e-12
Newton iteration   4energy = 8.623802355181667<A u 4 , A u 4 >_{-1}^0.5 =  8.517398348223816e-15
loadstep 44
Newton iteration   0energy = 8.619877920720509<A u 0 , A u 0 >_{-1}^0.5 =  0.004274620970054603
Newton iteration   1energy = 8.619868904667033<A u 1 , A u 1 >_{-1}^0.5 =  0.0001942491882238703
Newton iteration   2energy = 8.619868885790854<A u 2 , A u 2 >_{-1}^0.5 =  3.850411392703588e-07
Newton iteration   3energy = 8.619868885790778<A u 3 , A u 3 >_{-1}^0.5 =  4.240402111982851e-12
Newton iteration   4energy = 8.619868885790774<A u 4 , A u 4 >_{-1}^0.5 =  8.326519656718127e-15
loadstep 45
Newton iteration   0energy = 8.61592648119783<A u 0 , A u 0 >_{-1}^0.5 =  0.004204041915484391
Newton iteration   1energy = 8.615917756073554<A u 1 , A u 1 >_{-1}^0.5 =  0.0001819722236202162
Newton iteration   2energy = 8.61591773950842<A u 2 , A u 2 >_{-1}^0.5 =  3.3424495536253416e-07
Newton iteration   3energy = 8.615917739508363<A u 3 , A u 3 >_{-1}^0.5 =  3.1180198804711842e-12
Newton iteration   4energy = 8.615917739508362<A u 4 , A u 4 >_{-1}^0.5 =  8.729211484647772e-15
loadstep 46
Newton iteration   0energy = 8.611957945416394<A u 0 , A u 0 >_{-1}^0.5 =  0.0041365013915693
Newton iteration   1energy = 8.611949494472022<A u 1 , A u 1 >_{-1}^0.5 =  0.0001707627056471081
Newton iteration   2energy = 8.611949479885219<A u 2 , A u 2 >_{-1}^0.5 =  2.908968256549781e-07
Newton iteration   3energy = 8.611949479885176<A u 3 , A u 3 >_{-1}^0.5 =  2.3045214862315513e-12
Newton iteration   4energy = 8.611949479885176<A u 4 , A u 4 >_{-1}^0.5 =  9.476505735026269e-15
loadstep 47
Newton iteration   0energy = 8.60797284321524<A u 0 , A u 0 >_{-1}^0.5 =  0.004071817276964948
Newton iteration   1energy = 8.607964650961378<A u 1 , A u 1 >_{-1}^0.5 =  0.00016050915725754933
Newton iteration   2energy = 8.60796463807402<A u 2 , A u 2 >_{-1}^0.5 =  2.5381012040075324e-07
Newton iteration   3energy = 8.607964638073991<A u 3 , A u 3 >_{-1}^0.5 =  1.7112825412101932e-12
Newton iteration   4energy = 8.60796463807399<A u 4 , A u 4 >_{-1}^0.5 =  9.433321793785772e-15
loadstep 48
Newton iteration   0energy = 8.60397167462065<A u 0 , A u 0 >_{-1}^0.5 =  0.004009820960714137
Newton iteration   1energy = 8.603963726715204<A u 1 , A u 1 >_{-1}^0.5 =  0.0001511133171328678
Newton iteration   2energy = 8.603963715292728<A u 2 , A u 2 >_{-1}^0.5 =  2.220010106422092e-07
Newton iteration   3energy = 8.603963715292714<A u 3 , A u 3 >_{-1}^0.5 =  1.275556391486985e-12
Newton iteration   4energy = 8.603963715292702<A u 4 , A u 4 >_{-1}^0.5 =  9.686107577932144e-15
loadstep 49
Newton iteration   0energy = 8.599954912069549<A u 0 , A u 0 >_{-1}^0.5 =  0.003950356174564131
Newton iteration   1energy = 8.59994719521642<A u 1 , A u 1 >_{-1}^0.5 =  0.0001424884055043248
Newton iteration   2energy = 8.599947185060834<A u 2 , A u 2 >_{-1}^0.5 =  1.9465161594403441e-07
Newton iteration   3energy = 8.599947185060826<A u 3 , A u 3 >_{-1}^0.5 =  9.56721335441428e-13
Newton iteration   4energy = 8.599947185060817<A u 4 , A u 4 >_{-1}^0.5 =  1.01442216725829e-14

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.

In [10]:

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 += SymbolicEnergy (eps/2*grad(u)*grad(u)
                     + ((1-u*u)*(1-u*u))
                     + 0.5/dt*(u-gfuold)*(u-gfuold))

In [11]:

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=False)
t = 0

In [12]:

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

Newton iteration   0energy = 0.41450268691109665<A u 0 , A u 0 >_{-1}^0.5 =  0.2809232585283059
Newton iteration   1energy = 0.37690029538736747<A u 1 , A u 1 >_{-1}^0.5 =  0.019176192659565692
Newton iteration   2energy = 0.3767158234035433<A u 2 , A u 2 >_{-1}^0.5 =  9.630379284983538e-05
Newton iteration   3energy = 0.37671581876625276<A u 3 , A u 3 >_{-1}^0.5 =  2.50838578139849e-09
Newton iteration   4energy = 0.3767158187662527<A u 4 , A u 4 >_{-1}^0.5 =  2.9355370163738083e-16
t =  0.1
Newton iteration   0energy = 0.34388852354503213<A u 0 , A u 0 >_{-1}^0.5 =  0.23599828005106344
Newton iteration   1energy = 0.3171273710024063<A u 1 , A u 1 >_{-1}^0.5 =  0.013452151221307806
Newton iteration   2energy = 0.3170366836102077<A u 2 , A u 2 >_{-1}^0.5 =  4.664802779278174e-05
Newton iteration   3energy = 0.3170366825221795<A u 3 , A u 3 >_{-1}^0.5 =  5.77460941464541e-10
Newton iteration   4energy = 0.3170366825221793<A u 4 , A u 4 >_{-1}^0.5 =  3.163968816040173e-16
t =  0.2
Newton iteration   0energy = 0.2955290108422929<A u 0 , A u 0 >_{-1}^0.5 =  0.18333693294869569
Newton iteration   1energy = 0.27922135353137706<A u 1 , A u 1 >_{-1}^0.5 =  0.008020819298689669
Newton iteration   2energy = 0.2791891435391795<A u 2 , A u 2 >_{-1}^0.5 =  1.6287112762876847e-05
Newton iteration   3energy = 0.2791891434065441<A u 3 , A u 3 >_{-1}^0.5 =  6.891275544015452e-11
Newton iteration   4energy = 0.279189143406544<A u 4 , A u 4 >_{-1}^0.5 =  3.4938661083456126e-16
t =  0.30000000000000004
Newton iteration   0energy = 0.26701124384040154<A u 0 , A u 0 >_{-1}^0.5 =  0.1332937206028686
Newton iteration   1energy = 0.25831515743654343<A u 1 , A u 1 >_{-1}^0.5 =  0.004187655354629702
Newton iteration   2energy = 0.258306383137749<A u 2 , A u 2 >_{-1}^0.5 =  4.374151896800372e-06
Newton iteration   3energy = 0.2583063831281824<A u 3 , A u 3 >_{-1}^0.5 =  4.879052593501367e-12
Newton iteration   4energy = 0.2583063831281824<A u 4 , A u 4 >_{-1}^0.5 =  3.4662576970143404e-16
t =  0.4
Newton iteration   0energy = 0.25217846565739477<A u 0 , A u 0 >_{-1}^0.5 =  0.09212071263353547
Newton iteration   1energy = 0.24799617525728274<A u 1 , A u 1 >_{-1}^0.5 =  0.001980203210485691
Newton iteration   2energy = 0.24799421401922642<A u 2 , A u 2 >_{-1}^0.5 =  9.676166525426937e-07
Newton iteration   3energy = 0.24799421401875826<A u 3 , A u 3 >_{-1}^0.5 =  2.352999798024643e-13
Newton iteration   4energy = 0.24799421401875832<A u 4 , A u 4 >_{-1}^0.5 =  3.6774826879915607e-16
t =  0.5
Newton iteration   0energy = 0.24516863305276446<A u 0 , A u 0 >_{-1}^0.5 =  0.06141505016588461
Newton iteration   1energy = 0.24330052673100364<A u 1 , A u 1 >_{-1}^0.5 =  0.0008740389330635108
Newton iteration   2energy = 0.24330014470463096<A u 2 , A u 2 >_{-1}^0.5 =  1.8731959108252394e-07
Newton iteration   3energy = 0.24330014470461345<A u 3 , A u 3 >_{-1}^0.5 =  8.746323616126442e-15
t =  0.6
Newton iteration   0energy = 0.24207365834808142<A u 0 , A u 0 >_{-1}^0.5 =  0.03997539205980796
Newton iteration   1energy = 0.2412795131472567<A u 1 , A u 1 >_{-1}^0.5 =  0.0003689785123362593
Newton iteration   2energy = 0.24127944507060162<A u 2 , A u 2 >_{-1}^0.5 =  3.3363958742417324e-08
Newton iteration   3energy = 0.24127944507060106<A u 3 , A u 3 >_{-1}^0.5 =  4.786315775211166e-16
t =  0.7
Newton iteration   0energy = 0.24076734060641394<A u 0 , A u 0 >_{-1}^0.5 =  0.02564726714970522
Newton iteration   1energy = 0.24043973271028393<A u 1 , A u 1 >_{-1}^0.5 =  0.0001520242371548029
Newton iteration   2energy = 0.24043972115431173<A u 2 , A u 2 >_{-1}^0.5 =  5.730888743528835e-09
Newton iteration   3energy = 0.24043972115431161<A u 3 , A u 3 >_{-1}^0.5 =  3.9497354526053974e-16
t =  0.7999999999999999
Newton iteration   0energy = 0.2402303872880181<A u 0 , A u 0 >_{-1}^0.5 =  0.016353675259529705
Newton iteration   1energy = 0.24009700062875125<A u 1 , A u 1 >_{-1}^0.5 =  6.246094502087166e-05
Newton iteration   2energy = 0.24009699867804568<A u 2 , A u 2 >_{-1}^0.5 =  1.0198825510076212e-09
Newton iteration   3energy = 0.24009699867804568<A u 3 , A u 3 >_{-1}^0.5 =  3.910355678385764e-16
t =  0.8999999999999999
Newton iteration   0energy = 0.24001177430570772<A u 0 , A u 0 >_{-1}^0.5 =  0.01046198510439957
Newton iteration   1energy = 0.23995713746382227<A u 1 , A u 1 >_{-1}^0.5 =  2.6467926688190138e-05
Newton iteration   2energy = 0.23995713711354497<A u 2 , A u 2 >_{-1}^0.5 =  2.2098631704426873e-10
Newton iteration   3energy = 0.23995713711354494<A u 3 , A u 3 >_{-1}^0.5 =  4.2671806155941764e-16
t =  0.9999999999999999
Newton iteration   0energy = 0.2399217819098009<A u 0 , A u 0 >_{-1}^0.5 =  0.006812071313422244
Newton iteration   1energy = 0.2398986060461037<A u 1 , A u 1 >_{-1}^0.5 =  1.230212547904795e-05
Newton iteration   2energy = 0.2398986059704322<A u 2 , A u 2 >_{-1}^0.5 =  7.099126365185817e-11
Newton iteration   3energy = 0.23989860597043222<A u 3 , A u 3 >_{-1}^0.5 =  3.8095111854675177e-16
t =  1.0999999999999999
Newton iteration   0energy = 0.23988309433332944<A u 0 , A u 0 >_{-1}^0.5 =  0.004621770715225076
Newton iteration   1energy = 0.23987242312588983<A u 1 , A u 1 >_{-1}^0.5 =  6.7942416189496265e-06
Newton iteration   2energy = 0.23987242310280898<A u 2 , A u 2 >_{-1}^0.5 =  3.209524829098261e-11
Newton iteration   3energy = 0.23987242310280898<A u 3 , A u 3 >_{-1}^0.5 =  4.2018315330435895e-16
t =  1.2
Newton iteration   0energy = 0.23986480836454233<A u 0 , A u 0 >_{-1}^0.5 =  0.003368954272575993
Newton iteration   1energy = 0.23985913748728932<A u 1 , A u 1 >_{-1}^0.5 =  4.576099681476189e-06
Newton iteration   2energy = 0.2398591374768189<A u 2 , A u 2 >_{-1}^0.5 =  1.7623725590715833e-11
Newton iteration   3energy = 0.23985913747681892<A u 3 , A u 3 >_{-1}^0.5 =  4.270044197551166e-16
t =  1.3
Newton iteration   0energy = 0.23985470889450997<A u 0 , A u 0 >_{-1}^0.5 =  0.002692728902482192
Newton iteration   1energy = 0.23985108571269736<A u 1 , A u 1 >_{-1}^0.5 =  3.5610540851343554e-06
Newton iteration   2energy = 0.2398510857063568<A u 2 , A u 2 >_{-1}^0.5 =  1.0946944604129284e-11
Newton iteration   3energy = 0.23985108570635671<A u 3 , A u 3 >_{-1}^0.5 =  4.396452732575224e-16
t =  1.4000000000000001
Newton iteration   0energy = 0.23984799171546298<A u 0 , A u 0 >_{-1}^0.5 =  0.0023401679038810375
Newton iteration   1energy = 0.2398452549179071<A u 1 , A u 1 >_{-1}^0.5 =  3.005473339170729e-06
Newton iteration   2energy = 0.23984525491339054<A u 2 , A u 2 >_{-1}^0.5 =  7.442673310088071e-12
Newton iteration   3energy = 0.23984525491339057<A u 3 , A u 3 >_{-1}^0.5 =  4.491259343620475e-16
t =  1.5000000000000002
Newton iteration   0energy = 0.23984276557831818<A u 0 , A u 0 >_{-1}^0.5 =  0.0021500366176415617
Newton iteration   1energy = 0.23984045520885888<A u 1 , A u 1 >_{-1}^0.5 =  2.6479579130120444e-06
Newton iteration   2energy = 0.23984045520535296<A u 2 , A u 2 >_{-1}^0.5 =  5.422860528597392e-12
Newton iteration   3energy = 0.23984045520535302<A u 3 , A u 3 >_{-1}^0.5 =  4.1474306886148735e-16
t =  1.6000000000000003
Newton iteration   0energy = 0.23983827920581985<A u 0 , A u 0 >_{-1}^0.5 =  0.0020348851953489473
Newton iteration   1energy = 0.2398362095246352<A u 1 , A u 1 >_{-1}^0.5 =  2.3877185904425255e-06
Newton iteration   2energy = 0.2398362095217846<A u 2 , A u 2 >_{-1}^0.5 =  4.161521766436116e-12
Newton iteration   3energy = 0.2398362095217846<A u 3 , A u 3 >_{-1}^0.5 =  4.120990651108714e-16
t =  1.7000000000000004
Newton iteration   0energy = 0.2398342271470195<A u 0 , A u 0 >_{-1}^0.5 =  0.001953342985032429
Newton iteration   1energy = 0.23983231990182843<A u 1 , A u 1 >_{-1}^0.5 =  2.181270857293332e-06
Newton iteration   2energy = 0.23983231989944948<A u 2 , A u 2 >_{-1}^0.5 =  3.3146217660229777e-12
Newton iteration   3energy = 0.23983231989944945<A u 3 , A u 3 >_{-1}^0.5 =  4.3013449935758775e-16
t =  1.8000000000000005
Newton iteration   0energy = 0.23983047892544848<A u 0 , A u 0 >_{-1}^0.5 =  0.0018873221834727024
Newton iteration   1energy = 0.23982869834814605<A u 1 , A u 1 >_{-1}^0.5 =  2.0081124626660635e-06
Newton iteration   2energy = 0.23982869834612983<A u 2 , A u 2 >_{-1}^0.5 =  2.7089926940188982e-12
Newton iteration   3energy = 0.2398286983461298<A u 3 , A u 3 >_{-1}^0.5 =  4.136934679341362e-16
t =  1.9000000000000006
Newton iteration   0energy = 0.23982697342606712<A u 0 , A u 0 >_{-1}^0.5 =  0.0018291572337895208
Newton iteration   1energy = 0.23982530085281606<A u 1 , A u 1 >_{-1}^0.5 =  1.8577922988903242e-06
Newton iteration   2energy = 0.23982530085109044<A u 2 , A u 2 >_{-1}^0.5 =  2.253601788515063e-12
Newton iteration   3energy = 0.2398253008510904<A u 3 , A u 3 >_{-1}^0.5 =  4.161732600430683e-16
t =  2.0000000000000004
Newton iteration   0energy = 0.23982367762908566<A u 0 , A u 0 >_{-1}^0.5 =  0.0017755912541635457
Newton iteration   1energy = 0.23982210154298106<A u 1 , A u 1 >_{-1}^0.5 =  1.7245245368461245e-06
Newton iteration   2energy = 0.23982210154149403<A u 2 , A u 2 >_{-1}^0.5 =  1.8979705571118188e-12
Newton iteration   3energy = 0.23982210154149403<A u 3 , A u 3 >_{-1}^0.5 =  3.9958496757931224e-16
t =  2.1000000000000005
Newton iteration   0energy = 0.2398205703929944<A u 0 , A u 0 >_{-1}^0.5 =  0.0017251940955923843
Newton iteration   1energy = 0.23981908247720582<A u 1 , A u 1 >_{-1}^0.5 =  1.6048109259649143e-06
Newton iteration   2energy = 0.23981908247591813<A u 2 , A u 2 >_{-1}^0.5 =  1.6126836894271255e-12
Newton iteration   3energy = 0.2398190824759181<A u 3 , A u 3 >_{-1}^0.5 =  3.66015241147275e-16
t =  2.2000000000000006
Newton iteration   0energy = 0.2398176360054435<A u 0 , A u 0 >_{-1}^0.5 =  0.0016772924085901265
Newton iteration   1energy = 0.2398162295473618<A u 1 , A u 1 >_{-1}^0.5 =  1.4963465114958576e-06
Newton iteration   2energy = 0.2398162295462423<A u 2 , A u 2 >_{-1}^0.5 =  1.3792864549581004e-12
Newton iteration   3energy = 0.23981622954624232<A u 3 , A u 3 >_{-1}^0.5 =  4.0724372300964665e-16
t =  2.3000000000000007
Newton iteration   0energy = 0.23981486155942658<A u 0 , A u 0 >_{-1}^0.5 =  0.0016315314196973449
Newton iteration   1energy = 0.23981353078108197<A u 1 , A u 1 >_{-1}^0.5 =  1.3974937765274695e-06
Newton iteration   2energy = 0.23981353078010562<A u 2 , A u 2 >_{-1}^0.5 =  1.185661712735239e-12
Newton iteration   3energy = 0.2398135307801056<A u 3 , A u 3 >_{-1}^0.5 =  4.0692764831583306e-16
t =  2.400000000000001
Newton iteration   0energy = 0.23981223583993277<A u 0 , A u 0 >_{-1}^0.5 =  0.001587695231112611
Newton iteration   1energy = 0.23981097559818157<A u 1 , A u 1 >_{-1}^0.5 =  1.307016701083898e-06
Newton iteration   2energy = 0.23981097559732756<A u 2 , A u 2 >_{-1}^0.5 =  1.0234087151902238e-12
Newton iteration   3energy = 0.23981097559732759<A u 3 , A u 3 >_{-1}^0.5 =  4.2109059650142165e-16
t =  2.500000000000001
Newton iteration   0energy = 0.23980974881596231<A u 0 , A u 0 >_{-1}^0.5 =  0.001545632320221452
Newton iteration   1energy = 0.23980855445375673<A u 1 , A u 1 >_{-1}^0.5 =  1.2239382146600934e-06
Newton iteration   2energy = 0.23980855445300783<A u 2 , A u 2 >_{-1}^0.5 =  8.864704675716774e-13
Newton iteration   3energy = 0.2398085544530078<A u 3 , A u 3 >_{-1}^0.5 =  3.860503999338409e-16
t =  2.600000000000001
Newton iteration   0energy = 0.23980739138160234<A u 0 , A u 0 >_{-1}^0.5 =  0.0015052239652246484
Newton iteration   1energy = 0.2398062586435172<A u 1 , A u 1 >_{-1}^0.5 =  1.1474589658054772e-06
Newton iteration   2energy = 0.23980625864285895<A u 2 , A u 2 >_{-1}^0.5 =  7.700762845886604e-13
Newton iteration   3energy = 0.23980625864285893<A u 3 , A u 3 >_{-1}^0.5 =  4.3557538295205123e-16
t =  2.700000000000001
Newton iteration   0energy = 0.23980515520427065<A u 0 , A u 0 >_{-1}^0.5 =  0.0014663703545176434
Newton iteration   1energy = 0.23980408018121405<A u 1 , A u 1 >_{-1}^0.5 =  1.0769081798492949e-06
Newton iteration   2energy = 0.23980408018063423<A u 2 , A u 2 >_{-1}^0.5 =  6.70806280629591e-13
Newton iteration   3energy = 0.23980408018063418<A u 3 , A u 3 >_{-1}^0.5 =  4.1211995666328175e-16
t =  2.800000000000001
Newton iteration   0energy = 0.2398030326225266<A u 0 , A u 0 >_{-1}^0.5 =  0.0014289841085793756
Newton iteration   1energy = 0.23980201171105245<A u 1 , A u 1 >_{-1}^0.5 =  1.0117122443812797e-06
Newton iteration   2energy = 0.2398020117105407<A u 2 , A u 2 >_{-1}^0.5 =  5.857209839429618e-13
Newton iteration   3energy = 0.23980201171054066<A u 3 , A u 3 >_{-1}^0.5 =  4.136326079471177e-16
t =  2.9000000000000012
Newton iteration   0energy = 0.23980101656930852<A u 0 , A u 0 >_{-1}^0.5 =  0.0013929869965262228
Newton iteration   1energy = 0.23980004643919056<A u 1 , A u 1 >_{-1}^0.5 =  9.513736147237864e-07
Newton iteration   2energy = 0.23980004643873795<A u 2 , A u 2 >_{-1}^0.5 =  5.12637238731772e-13
Newton iteration   3energy = 0.23980004643873803<A u 3 , A u 3 >_{-1}^0.5 =  4.0818478086929167e-16
t =  3.0000000000000013
Newton iteration   0energy = 0.23979910050996395<A u 0 , A u 0 >_{-1}^0.5 =  0.0013583080881265145
Newton iteration   1energy = 0.23979817807707918<A u 1 , A u 1 >_{-1}^0.5 =  8.954560268805773e-07
Newton iteration   2energy = 0.23979817807667816<A u 2 , A u 2 >_{-1}^0.5 =  4.496599978112209e-13
Newton iteration   3energy = 0.2397981780766781<A u 3 , A u 3 >_{-1}^0.5 =  3.7320188102935894e-16
t =  3.1000000000000014
Newton iteration   0energy = 0.23979727839003487<A u 0 , A u 0 >_{-1}^0.5 =  0.001324882589652414
Newton iteration   1energy = 0.23979640079304174<A u 1 , A u 1 >_{-1}^0.5 =  8.435737332075263e-07
Newton iteration   2energy = 0.23979640079268588<A u 2 , A u 2 >_{-1}^0.5 =  3.9516854540745673e-13
Newton iteration   3energy = 0.23979640079268588<A u 3 , A u 3 >_{-1}^0.5 =  3.7393791096780473e-16
t =  3.2000000000000015
Newton iteration   0energy = 0.23979554459015517<A u 0 , A u 0 >_{-1}^0.5 =  0.0012926510309648675
Newton iteration   1energy = 0.23979470917010964<A u 1 , A u 1 >_{-1}^0.5 =  7.953834056061581e-07
Newton iteration   2energy = 0.23979470916979337<A u 2 , A u 2 >_{-1}^0.5 =  3.480245574719867e-13
Newton iteration   3energy = 0.23979470916979337<A u 3 , A u 3 >_{-1}^0.5 =  3.872132135793069e-16
t =  3.3000000000000016
Newton iteration   0energy = 0.2397938938865219<A u 0 , A u 0 >_{-1}^0.5 =  0.001261558649417167
Newton iteration   1energy = 0.2397930981688792<A u 1 , A u 1 >_{-1}^0.5 =  7.505778682442833e-07
Newton iteration   2energy = 0.23979309816859737<A u 2 , A u 2 >_{-1}^0.5 =  3.070918168492527e-13
Newton iteration   3energy = 0.2397930981685974<A u 3 , A u 3 >_{-1}^0.5 =  3.7432778189623423e-16
t =  3.4000000000000017
Newton iteration   0energy = 0.2397923214159341<A u 0 , A u 0 >_{-1}^0.5 =  0.001231554894418221
Newton iteration   1energy = 0.23979156309455196<A u 1 , A u 1 >_{-1}^0.5 =  7.088811244437789e-07
Newton iteration   2energy = 0.23979156309430072<A u 2 , A u 2 >_{-1}^0.5 =  2.7146435119184975e-13
Newton iteration   3energy = 0.2397915630943007<A u 3 , A u 3 >_{-1}^0.5 =  4.055972384702122e-16
t =  3.5000000000000018
Newton iteration   0energy = 0.239790822644684<A u 0 , A u 0 >_{-1}^0.5 =  0.0012025930123929097
Newton iteration   1energy = 0.2397900995675406<A u 1 , A u 1 >_{-1}^0.5 =  6.700443260192641e-07
Newton iteration   2energy = 0.23979009956731617<A u 2 , A u 2 >_{-1}^0.5 =  2.4039198923922215e-13
Newton iteration   3energy = 0.2397900995673162<A u 3 , A u 3 >_{-1}^0.5 =  4.1530851920836197e-16
t =  3.600000000000002
Newton iteration   0energy = 0.2397893933407546<A u 0 , A u 0 >_{-1}^0.5 =  0.0011746296892646987
Newton iteration   1energy = 0.2397887034971503<A u 1 , A u 1 >_{-1}^0.5 =  6.338424467591721e-07
Newton iteration   2energy = 0.23978870349694945<A u 2 , A u 2 >_{-1}^0.5 =  2.1332234861683952e-13
Newton iteration   3energy = 0.23978870349694942<A u 3 , A u 3 >_{-1}^0.5 =  3.994075992214168e-16
t =  3.700000000000002
Newton iteration   0energy = 0.23978802954889125<A u 0 , A u 0 >_{-1}^0.5 =  0.0011476247365417888
Newton iteration   1energy = 0.23978737105794926<A u 1 , A u 1 >_{-1}^0.5 =  6.00071494532039e-07
Newton iteration   2energy = 0.2397873710577693<A u 2 , A u 2 >_{-1}^0.5 =  1.8963750662706926e-13
Newton iteration   3energy = 0.23978737105776934<A u 3 , A u 3 >_{-1}^0.5 =  3.9508651477264857e-16
t =  3.800000000000002
Newton iteration   0energy = 0.2397867275681889<A u 0 , A u 0 >_{-1}^0.5 =  0.001121540812005155
Newton iteration   1energy = 0.23978609866849762<A u 1 , A u 1 >_{-1}^0.5 =  5.685461452199974e-07
Newton iteration   2energy = 0.239786098668336<A u 2 , A u 2 >_{-1}^0.5 =  1.6881169362487337e-13
Newton iteration   3energy = 0.23978609866833606<A u 3 , A u 3 >_{-1}^0.5 =  3.7362237903421027e-16
t =  3.900000000000002
Newton iteration   0energy = 0.23978548393189394<A u 0 , A u 0 >_{-1}^0.5 =  0.001096343168856977
Newton iteration   1energy = 0.23978488297215855<A u 1 , A u 1 >_{-1}^0.5 =  5.390977122982297e-07
Newton iteration   2energy = 0.2397848829720133<A u 2 , A u 2 >_{-1}^0.5 =  1.506206667808283e-13
Newton iteration   3energy = 0.23978488297201328<A u 3 , A u 3 >_{-1}^0.5 =  4.0256707927596727e-16
t =  4.000000000000002
Newton iteration   0energy = 0.2397842953891668<A u 0 , A u 0 >_{-1}^0.5 =  0.001071999428939726
Newton iteration   1energy = 0.2397837208197555<A u 1 , A u 1 >_{-1}^0.5 =  5.115723902938338e-07
Newton iteration   2energy = 0.23978372081962457<A u 2 , A u 2 >_{-1}^0.5 =  1.3461240330667862e-13
Newton iteration   3energy = 0.23978372081962462<A u 3 , A u 3 >_{-1}^0.5 =  4.22810320364615e-16
t =  4.100000000000001
Newton iteration   0energy = 0.23978315888858148<A u 0 , A u 0 >_{-1}^0.5 =  0.0010484793767550234
Newton iteration   1energy = 0.23978260925386716<A u 1 , A u 1 >_{-1}^0.5 =  4.858297225032861e-07
Newton iteration   2energy = 0.23978260925374922<A u 2 , A u 2 >_{-1}^0.5 =  1.2049305730754915e-13
Newton iteration   3energy = 0.23978260925374922<A u 3 , A u 3 >_{-1}^0.5 =  4.2049767882605063e-16
t =  4.200000000000001
Newton iteration   0energy = 0.23978207156317277<A u 0 , A u 0 >_{-1}^0.5 =  0.0010257547717514503
Newton iteration   1energy = 0.23978154549458386<A u 1 , A u 1 >_{-1}^0.5 =  4.6174125702286747e-07
Newton iteration   2energy = 0.23978154549447736<A u 2 , A u 2 >_{-1}^0.5 =  1.0804519197732318e-13
Newton iteration   3energy = 0.23978154549447733<A u 3 , A u 3 >_{-1}^0.5 =  3.75485072305106e-16
t =  4.300000000000001
Newton iteration   0energy = 0.23978103071686038<A u 0 , A u 0 >_{-1}^0.5 =  0.0010037991768567201
Newton iteration   1energy = 0.23978052692656368<A u 1 , A u 1 >_{-1}^0.5 =  4.3918936193603866e-07
Newton iteration   2energy = 0.2397805269264672<A u 2 , A u 2 >_{-1}^0.5 =  9.708485837881644e-14
t =  4.4
Newton iteration   0energy = 0.23978003381210533<A u 0 , A u 0 >_{-1}^0.5 =  0.0009825878015857251
Newton iteration   1energy = 0.23977955108725418<A u 1 , A u 1 >_{-1}^0.5 =  4.180661760124394e-07
Newton iteration   2energy = 0.23977955108716673<A u 2 , A u 2 >_{-1}^0.5 =  8.732008667811928e-14
t =  4.5
Newton iteration   0energy = 0.23977907845866553<A u 0 , A u 0 >_{-1}^0.5 =  0.0009620973583148351
Newton iteration   1energy = 0.2397786156561545<A u 1 , A u 1 >_{-1}^0.5 =  3.9827267609246794e-07
Newton iteration   2energy = 0.23977861565607522<A u 2 , A u 2 >_{-1}^0.5 =  7.871313441500204e-14
t =  4.6
Newton iteration   0energy = 0.23977816240333752<A u 0 , A u 0 >_{-1}^0.5 =  0.0009423059305070753
Newton iteration   1energy = 0.23977771844501206<A u 1 , A u 1 >_{-1}^0.5 =  3.797178460263959e-07
Newton iteration   2energy = 0.2397777184449398<A u 2 , A u 2 >_{-1}^0.5 =  7.107647461255042e-14
t =  4.699999999999999
Newton iteration   0energy = 0.23977728352058292<A u 0 , A u 0 >_{-1}^0.5 =  0.0009231928518233563
Newton iteration   1energy = 0.23977685738885424<A u 1 , A u 1 >_{-1}^0.5 =  3.6231793361432817e-07
Newton iteration   2energy = 0.23977685738878857<A u 2 , A u 2 >_{-1}^0.5 =  6.426868881058818e-14
t =  4.799999999999999
Newton iteration   0energy = 0.2397764398039481<A u 0 , A u 0 >_{-1}^0.5 =  0.0009047385951755221
Newton iteration   1energy = 0.23977603053777619<A u 1 , A u 1 >_{-1}^0.5 =  3.459957855427896e-07
Newton iteration   2energy = 0.23977603053771646<A u 2 , A u 2 >_{-1}^0.5 =  5.827355843816445e-14
t =  4.899999999999999
Newton iteration   0energy = 0.23977562935819674<A u 0 , A u 0 >_{-1}^0.5 =  0.0008869246708769174
Newton iteration   1energy = 0.23977523604940246<A u 1 , A u 1 >_{-1}^0.5 =  3.306802497027191e-07
Newton iteration   2energy = 0.2397752360493478<A u 2 , A u 2 >_{-1}^0.5 =  5.288358902541465e-14
t =  4.999999999999998