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 ngsolve import *
import netgen.gui

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 =  0.13255957337312613
Newton iteration   1energy = -0.008785676900152117<A u 1 , A u 1 >_{-1}^0.5 =  1.1107598189167146e-05
Newton iteration   2energy = -0.008785676961841486<A u 2 , A u 2 >_{-1}^0.5 =  2.807517603399629e-13
Newton iteration   3energy = -0.008785676961841488<A u 3 , A u 3 >_{-1}^0.5 =  3.3304484794486327e-17
energy =  -0.008785676961841488

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 =  0.13255957337312615
Newton iteration  1
Energy:  -0.008785676900152117
err =  1.1107598189171795e-05
Newton iteration  2
Energy:  -0.008785676961841486
err =  2.80747021639635e-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 = VectorH1(mesh, order=2, dirichlet="left")
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

factor = Parameter(1.0)

a = BilinearForm(V, symmetric=True)
a += Variation(  NeoHook (C).Compile() * dx
                -factor * (InnerProduct(force,u) ).Compile() * dx)

We want to solve the minimization problem for $\gamma = 5$. Due to the high nonlinearity in the problem, the Newton iteration will not convergence with any initial guess. We approach the case $\gamma = 5$ by solving problems with $\gamma = i/10$ for $i=1,..,50$ and taking the solution of the previous problem as an initial guess.

[10]:

gfu = GridFunction(V)
gfu.vec[:] = 0

Draw (gfu, mesh, "u")
SetVisualization (deformation=True)

res = gfu.vec.CreateVector()
du = gfu.vec.CreateVector()

for loadstep in range(50):
    print ("loadstep", loadstep)
    factor.Set ((loadstep+1)/10)
    SolveNonlinearMinProblem(a,gfu)
    Redraw()

loadstep 0
Newton iteration   0energy = 8.749999999999998<A u 0 , A u 0 >_{-1}^0.5 =  0.01667849444523386
Newton iteration   1energy = 8.750132591453418<A u 1 , A u 1 >_{-1}^0.5 =  0.023333560516275564
Newton iteration   2energy = 8.74986114895739<A u 2 , A u 2 >_{-1}^0.5 =  0.00010232534273250714
Newton iteration   3energy = 8.749861143722152<A u 3 , A u 3 >_{-1}^0.5 =  5.1484913353753e-08
Newton iteration   4energy = 8.749861143722145<A u 4 , A u 4 >_{-1}^0.5 =  2.734694947345281e-13
Newton iteration   5energy = 8.749861143722145<A u 5 , A u 5 >_{-1}^0.5 =  9.247273008118622e-16
loadstep 1
Newton iteration   0energy = 8.74958388866222<A u 0 , A u 0 >_{-1}^0.5 =  0.016596172790229985
Newton iteration   1energy = 8.749710096254224<A u 1 , A u 1 >_{-1}^0.5 =  0.022958523347670023
Newton iteration   2energy = 8.749447295506688<A u 2 , A u 2 >_{-1}^0.5 =  0.00014215759220200522
Newton iteration   3energy = 8.749447285508145<A u 3 , A u 3 >_{-1}^0.5 =  2.325574122106694e-06
Newton iteration   4energy = 8.749447285505438<A u 4 , A u 4 >_{-1}^0.5 =  6.329020240483118e-10
Newton iteration   5energy = 8.749447285505443<A u 5 , A u 5 >_{-1}^0.5 =  1.6967201531172987e-15
loadstep 2
Newton iteration   0energy = 8.748898140386652<A u 0 , A u 0 >_{-1}^0.5 =  0.016356983393359056
Newton iteration   1energy = 8.749005259290692<A u 1 , A u 1 >_{-1}^0.5 =  0.021892894044873473
Newton iteration   2energy = 8.748766252622183<A u 2 , A u 2 >_{-1}^0.5 =  0.000208222077261969
Newton iteration   3energy = 8.748766231266895<A u 3 , A u 3 >_{-1}^0.5 =  4.890915769550513e-06
Newton iteration   4energy = 8.748766231254931<A u 4 , A u 4 >_{-1}^0.5 =  2.665249442467225e-09
Newton iteration   5energy = 8.748766231254935<A u 5 , A u 5 >_{-1}^0.5 =  2.2360660986729218e-15
loadstep 3
Newton iteration   0energy = 8.747955292062146<A u 0 , A u 0 >_{-1}^0.5 =  0.0159820733778304
Newton iteration   1energy = 8.74803542180493<A u 1 , A u 1 >_{-1}^0.5 =  0.020292399713242315
Newton iteration   2energy = 8.747830036432735<A u 2 , A u 2 >_{-1}^0.5 =  0.00026380399717611486
Newton iteration   3energy = 8.747830002128063<A u 3 , A u 3 >_{-1}^0.5 =  7.24731483392178e-06
Newton iteration   4energy = 8.747830002101788<A u 4 , A u 4 >_{-1}^0.5 =  4.744152101219956e-09
Newton iteration   5energy = 8.747830002101791<A u 5 , A u 5 >_{-1}^0.5 =  3.834758384860599e-15
loadstep 4
Newton iteration   0energy = 8.746771010509391<A u 0 , A u 0 >_{-1}^0.5 =  0.015501067150346784
Newton iteration   1energy = 8.746821856285143<A u 1 , A u 1 >_{-1}^0.5 =  0.01835991696222946
Newton iteration   2energy = 8.746653679174742<A u 2 , A u 2 >_{-1}^0.5 =  0.00029906276685253894
Newton iteration   3energy = 8.746653634982097<A u 3 , A u 3 >_{-1}^0.5 =  8.720360402018377e-06
Newton iteration   4energy = 8.746653634944062<A u 4 , A u 4 >_{-1}^0.5 =  5.243058230169497e-09
Newton iteration   5energy = 8.746653634944062<A u 5 , A u 5 >_{-1}^0.5 =  4.57624018420641e-15
loadstep 5
Newton iteration   0energy = 8.745362710869802<A u 0 , A u 0 >_{-1}^0.5 =  0.01494623240146039
Newton iteration   1energy = 8.745386462940143<A u 1 , A u 1 >_{-1}^0.5 =  0.016292996834189524
Newton iteration   2energy = 8.74525397874261<A u 2 , A u 2 >_{-1}^0.5 =  0.00031320373730301293
Newton iteration   3energy = 8.745253930142317<A u 3 , A u 3 >_{-1}^0.5 =  9.042918405812291e-06
Newton iteration   4energy = 8.745253930101422<A u 4 , A u 4 >_{-1}^0.5 =  4.2170747228818445e-09
Newton iteration   5energy = 8.74525393010142<A u 5 , A u 5 >_{-1}^0.5 =  4.4934195509996876e-15
loadstep 6
Newton iteration   0energy = 8.743748369013607<A u 0 , A u 0 >_{-1}^0.5 =  0.014347713355602996
Newton iteration   1energy = 8.743749760522784<A u 1 , A u 1 >_{-1}^0.5 =  0.014249104212014871
Newton iteration   2energy = 8.74364839737191<A u 2 , A u 2 >_{-1}^0.5 =  0.00030934606309694624
Newton iteration   3energy = 8.743648349846454<A u 3 , A u 3 >_{-1}^0.5 =  8.384888859530472e-06
Newton iteration   4energy = 8.743648349811291<A u 4 , A u 4 >_{-1}^0.5 =  2.6940148501867647e-09
Newton iteration   5energy =  8.7436483498113<A u 5 , A u 5 >_{-1}^0.5 =  4.577084252820437e-15
loadstep 7
Newton iteration   0energy = 8.741945639262823<A u 0 , A u 0 >_{-1}^0.5 =  0.013730721530226796
Newton iteration   1energy = 8.741930180324744<A u 1 , A u 1 >_{-1}^0.5 =  0.012333575792311013
Newton iteration   2energy = 8.741854213024633<A u 2 , A u 2 >_{-1}^0.5 =  0.0002921321706594058
Newton iteration   3energy = 8.74185417055718<A u 3 , A u 3 >_{-1}^0.5 =  7.130071890016913e-06
Newton iteration   4energy = 8.741854170531763<A u 4 , A u 4 >_{-1}^0.5 =  1.4434881033484502e-09
Newton iteration   5energy = 8.74185417053177<A u 5 , A u 5 >_{-1}^0.5 =  5.262958683534736e-15
loadstep 8
Newton iteration   0energy = 8.739971280231286<A u 0 , A u 0 >_{-1}^0.5 =  0.013114572255962758
Newton iteration   1energy = 8.73994411367146<A u 1 , A u 1 >_{-1}^0.5 =  0.010604017031859161
Newton iteration   2energy = 8.739887940284762<A u 2 , A u 2 >_{-1}^0.5 =  0.00026632144941452456
Newton iteration   3energy = 8.73988790493644<A u 3 , A u 3 >_{-1}^0.5 =  5.662818925120992e-06
Newton iteration   4energy = 8.739887904920405<A u 4 , A u 4 >_{-1}^0.5 =  6.720247861727614e-10
Newton iteration   5energy = 8.739887904920398<A u 5 , A u 5 >_{-1}^0.5 =  5.5915825831605006e-15
loadstep 9
Newton iteration   0energy = 8.73784083403977<A u 0 , A u 0 >_{-1}^0.5 =  0.0125129674198164
Newton iteration   1energy = 8.737806204166365<A u 1 , A u 1 >_{-1}^0.5 =  0.009082089187694535
Newton iteration   2energy = 8.73776498517593<A u 2 , A u 2 >_{-1}^0.5 =  0.00023604345317251825
Newton iteration   3energy = 8.73776495737735<A u 3 , A u 3 >_{-1}^0.5 =  4.2585795846192444e-06
Newton iteration   4energy = 8.737764957368277<A u 4 , A u 4 >_{-1}^0.5 =  2.778882710277497e-10
Newton iteration   5energy = 8.73776495736828<A u 5 , A u 5 >_{-1}^0.5 =  6.192838256475658e-15
loadstep 10
Newton iteration   0energy = 8.735568488681253<A u 0 , A u 0 >_{-1}^0.5 =  0.011934895047581391
Newton iteration   1energy = 8.735529626298675<A u 1 , A u 1 >_{-1}^0.5 =  0.007766086303887108
Newton iteration   2energy = 8.735499478251771<A u 2 , A u 2 >_{-1}^0.5 =  0.00020450110958987553
Newton iteration   3energy = 8.735499457369832<A u 3 , A u 3 >_{-1}^0.5 =  3.0649641069813007e-06
Newton iteration   4energy = 8.73549945736514<A u 4 , A u 4 >_{-1}^0.5 =  1.0317259419213303e-10
Newton iteration   5energy = 8.735499457365144<A u 5 , A u 5 >_{-1}^0.5 =  6.1435177694761e-15
loadstep 11
Newton iteration   0energy = 8.733167060537248<A u 0 , A u 0 >_{-1}^0.5 =  0.011385697188469452
Newton iteration   1energy = 8.733126283518718<A u 1 , A u 1 >_{-1}^0.5 =  0.006641150145740644
Newton iteration   2energy = 8.733104230848486<A u 2 , A u 2 >_{-1}^0.5 =  0.00017394439104879262
Newton iteration   3energy = 8.733104215732606<A u 3 , A u 3 >_{-1}^0.5 =  2.129411853086759e-06
Newton iteration   4energy = 8.733104215730345<A u 4 , A u 4 >_{-1}^0.5 =  3.432558772877624e-11
Newton iteration   5energy = 8.733104215730338<A u 5 , A u 5 >_{-1}^0.5 =  8.094250286114186e-15
loadstep 12
Newton iteration   0energy = 8.730648048952677<A u 0 , A u 0 >_{-1}^0.5 =  0.01086806034480337
Newton iteration   1energy = 8.730606940950912<A u 1 , A u 1 >_{-1}^0.5 =  0.005686383297879251
Newton iteration   2energy = 8.730590769249666<A u 2 , A u 2 >_{-1}^0.5 =  0.00014578425109884895
Newton iteration   3energy = 8.730590758628075<A u 3 , A u 3 >_{-1}^0.5 =  1.4384336718370366e-06
Newton iteration   4energy = 8.730590758627047<A u 4 , A u 4 >_{-1}^0.5 =  9.987280236896591e-12
Newton iteration   5energy = 8.730590758627043<A u 5 , A u 5 >_{-1}^0.5 =  7.875973513059426e-15
loadstep 13
Newton iteration   0energy = 8.72802172950889<A u 0 , A u 0 >_{-1}^0.5 =  0.01038282956646614
Newton iteration   1energy = 8.727981322866317<A u 1 , A u 1 >_{-1}^0.5 =  0.004879270983755723
Newton iteration   2energy = 8.727969413471362<A u 2 , A u 2 >_{-1}^0.5 =  0.00012076411719468836
Newton iteration   3energy = 8.727969406181067<A u 3 , A u 3 >_{-1}^0.5 =  9.505522832199866e-07
Newton iteration   4energy = 8.727969406180618<A u 4 , A u 4 >_{-1}^0.5 =  2.327209811059151e-12
Newton iteration   5energy = 8.727969406180616<A u 5 , A u 5 >_{-1}^0.5 =  8.366395854739112e-15
loadstep 14
Newton iteration   0energy = 8.725297264364867<A u 0 , A u 0 >_{-1}^0.5 =  0.009929630827329049
Newton iteration   1energy = 8.725258196156323<A u 1 , A u 1 >_{-1}^0.5 =  0.004198149297602768
Newton iteration   2energy = 8.72524937798614<A u 2 , A u 2 >_{-1}^0.5 =  9.914019138175112e-05
Newton iteration   3energy = 8.725249373072183<A u 3 , A u 3 >_{-1}^0.5 =  6.177438537851986e-07
Newton iteration   4energy = 8.725249373071996<A u 4 , A u 4 >_{-1}^0.5 =  2.711757828663729e-13
Newton iteration   5energy = 8.72524937307199<A u 5 , A u 5 >_{-1}^0.5 =  8.086506222001764e-15
loadstep 15
Newton iteration   0energy = 8.722482816430372<A u 0 , A u 0 >_{-1}^0.5 =  0.009507324233198547
Newton iteration   1energy = 8.72244544987052<A u 1 , A u 1 >_{-1}^0.5 =  0.0036234023936600923
Newton iteration   2energy = 8.722438879876965<A u 2 , A u 2 >_{-1}^0.5 =  8.084368163731423e-05
Newton iteration   3energy = 8.722438876609113<A u 3 , A u 3 >_{-1}^0.5 =  3.966066049271973e-07
Newton iteration   4energy = 8.722438876609038<A u 4 , A u 4 >_{-1}^0.5 =  1.5278000509753718e-13
Newton iteration   5energy = 8.722438876609038<A u 5 , A u 5 >_{-1}^0.5 =  8.08115538661459e-15
loadstep 16
Newton iteration   0energy = 8.719585659767372<A u 0 , A u 0 >_{-1}^0.5 =  0.009114322388199479
Newton iteration   1energy = 8.71955017364448<A u 1 , A u 1 >_{-1}^0.5 =  0.00313789936072171
Newton iteration   2energy = 8.719545245704236<A u 2 , A u 2 >_{-1}^0.5 =  6.561232996881842e-05
Newton iteration   3energy = 8.71954524355164<A u 3 , A u 3 >_{-1}^0.5 =  2.52533093944595e-07
Newton iteration   4energy = 8.719545243551611<A u 4 , A u 4 >_{-1}^0.5 =  1.5033040334992298e-13
Newton iteration   5energy = 8.719545243551606<A u 5 , A u 5 >_{-1}^0.5 =  8.051020566432653e-15
loadstep 17
Newton iteration   0energy = 8.71661228222845<A u 0 , A u 0 >_{-1}^0.5 =  0.008748807494135346
Newton iteration   1energy = 8.716578734485331<A u 1 , A u 1 >_{-1}^0.5 =  0.0027270119708742158
Newton iteration   2energy = 8.716575012244606<A u 2 , A u 2 >_{-1}^0.5 =  5.308808014174092e-05
Newton iteration   3energy = 8.716575010835323<A u 3 , A u 3 >_{-1}^0.5 =  1.599966209600588e-07
Newton iteration   4energy = 8.716575010835307<A u 4 , A u 4 >_{-1}^0.5 =  9.245681199048317e-14
loadstep 18
Newton iteration   0energy = 8.713568478570464<A u 0 , A u 0 >_{-1}^0.5 =  0.008408875280311978
Newton iteration   1energy = 8.713536850394119<A u 1 , A u 1 >_{-1}^0.5 =  0.0023784245288753533
Newton iteration   2energy = 8.713534018737166<A u 2 , A u 2 >_{-1}^0.5 =  4.288364628582515e-05
Newton iteration   3energy = 8.71353401781759<A u 3 , A u 3 >_{-1}^0.5 =  1.011410754526628e-07
Newton iteration   4energy = 8.71353401781757<A u 4 , A u 4 >_{-1}^0.5 =  4.8832450515969854e-14
loadstep 19
Newton iteration   0energy = 8.710459433603457<A u 0 , A u 0 >_{-1}^0.5 =  0.008092627477923432
Newton iteration   1energy = 8.710429659421148<A u 1 , A u 1 >_{-1}^0.5 =  0.002081858695703015
Newton iteration   2energy = 8.710427489790115<A u 2 , A u 2 >_{-1}^0.5 =  3.462374278328321e-05
Newton iteration   3energy = 8.71042748919065<A u 3 , A u 3 >_{-1}^0.5 =  6.393601985494953e-08
Newton iteration   4energy = 8.710427489190652<A u 4 , A u 4 >_{-1}^0.5 =  2.505219063009473e-14
loadstep 20
Newton iteration   0energy = 8.707289795678951<A u 0 , A u 0 >_{-1}^0.5 =  0.007798228855193676
Newton iteration   1energy = 8.707261783232546<A u 1 , A u 1 >_{-1}^0.5 =  0.001828780371476849
Newton iteration   2energy = 8.707260108985894<A u 2 , A u 2 >_{-1}^0.5 =  2.796754346607161e-05
Newton iteration   3energy = 8.707260108594774<A u 3 , A u 3 >_{-1}^0.5 =  4.0490580515779663e-08
Newton iteration   4energy = 8.707260108594774<A u 4 , A u 4 >_{-1}^0.5 =  1.4094429423583045e-14
loadstep 21
Newton iteration   0energy = 8.704063741206715<A u 0 , A u 0 >_{-1}^0.5 =  0.007523940275902371
Newton iteration   1energy = 8.704037384768432<A u 1 , A u 1 >_{-1}^0.5 =  0.0016121223324776503
Newton iteration   2energy = 8.704036083705963<A u 2 , A u 2 >_{-1}^0.5 =  2.2618373003579114e-05
Newton iteration   3energy = 8.70403608345015<A u 3 , A u 3 >_{-1}^0.5 =  2.57264200426525e-08
Newton iteration   4energy = 8.704036083450148<A u 4 , A u 4 >_{-1}^0.5 =  1.025503816822179e-14
loadstep 22
Newton iteration   0energy = 8.700785031058096<A u 0 , A u 0 >_{-1}^0.5 =  0.007268135817663741
Newton iteration   1energy = 8.700760219966702<A u 1 , A u 1 >_{-1}^0.5 =  0.0014260372524824256
Newton iteration   2energy = 8.700759201928879<A u 2 , A u 2 >_{-1}^0.5 =  1.8325434222126603e-05
Newton iteration   3energy = 8.700759201760949<A u 3 , A u 3 >_{-1}^0.5 =  1.6417609086959264e-08
Newton iteration   4energy = 8.70075920176095<A u 4 , A u 4 >_{-1}^0.5 =  9.90254098112901e-15
loadstep 23
Newton iteration   0energy = 8.697457059755887<A u 0 , A u 0 >_{-1}^0.5 =  0.007029309502644591
Newton iteration   1energy = 8.697433683793736<A u 1 , A u 1 >_{-1}^0.5 =  0.0012656853969031537
Newton iteration   2energy = 8.69743288184044<A u 2 , A u 2 >_{-1}^0.5 =  1.4881058448227105e-05
Newton iteration   3energy = 8.69743288172972<A u 3 , A u 3 >_{-1}^0.5 =  1.05321833182215e-08
Newton iteration   4energy = 8.697432881729718<A u 4 , A u 4 >_{-1}^0.5 =  9.588291840221449e-15
loadstep 24
Newton iteration   0energy = 8.694082898322856<A u 0 , A u 0 >_{-1}^0.5 =  0.006806075429231539
Newton iteration   1energy = 8.694060850980257<A u 1 , A u 1 >_{-1}^0.5 =  0.001127056033072989
Newton iteration   2energy = 8.69406021509277<A u 2 , A u 2 >_{-1}^0.5 =  1.2115810532600391e-05
Newton iteration   3energy = 8.694060215019382<A u 3 , A u 3 >_{-1}^0.5 =  6.796446181591913e-09
Newton iteration   4energy = 8.694060215019384<A u 4 , A u 4 >_{-1}^0.5 =  1.0653141855886985e-14
loadstep 25
Newton iteration   0energy = 8.690665331596561<A u 0 , A u 0 >_{-1}^0.5 =  0.006597163862847386
Newton iteration   1energy = 8.690644511937402<A u 1 , A u 1 >_{-1}^0.5 =  0.001006819235980881
Newton iteration   2energy = 8.69064400450148<A u 2 , A u 2 >_{-1}^0.5 =  9.892890798941409e-06
Newton iteration   3energy = 8.690644004452542<A u 3 , A u 3 >_{-1}^0.5 =  4.413672136741346e-09
Newton iteration   4energy = 8.690644004452544<A u 4 , A u 4 >_{-1}^0.5 =  8.706654914525168e-15
loadstep 26
Newton iteration   0energy = 8.687206890738581<A u 0 , A u 0 >_{-1}^0.5 =  0.00640141499389159
Newton iteration   1energy = 8.68718720435108<A u 1 , A u 1 >_{-1}^0.5 =  0.0009022039607243906
Newton iteration   2energy = 8.687186796900923<A u 2 , A u 2 >_{-1}^0.5 =  8.102650665520232e-06
Newton iteration   3energy = 8.68718679686809<A u 3 , A u 3 >_{-1}^0.5 =  2.8854231418118854e-09
Newton iteration   4energy = 8.687186796868094<A u 4 , A u 4 >_{-1}^0.5 =  9.50793693421982e-15
loadstep 27
Newton iteration   0energy = 8.683709881582512<A u 0 , A u 0 >_{-1}^0.5 =  0.006217771488356258
Newton iteration   1energy = 8.683691240941258<A u 1 , A u 1 >_{-1}^0.5 =  0.0008108982398810496
Newton iteration   2energy = 8.683690911799431<A u 2 , A u 2 >_{-1}^0.5 =  6.6576303877916165e-06
Newton iteration   3energy = 8.683690911777271<A u 3 , A u 3 >_{-1}^0.5 =  1.8993156161478608e-09
Newton iteration   4energy = 8.683690911777267<A u 4 , A u 4 >_{-1}^0.5 =  9.362268095886705e-15
loadstep 28
Newton iteration   0energy = 8.680176409385007<A u 0 , A u 0 >_{-1}^0.5 =  0.006045270560225572
Newton iteration   1energy = 8.680158733842598<A u 1 , A u 1 >_{-1}^0.5 =  0.0007309677194524074
Newton iteration   2energy = 8.680158466399725<A u 2 , A u 2 >_{-1}^0.5 =  5.488278554618818e-06
Newton iteration   3energy = 8.680158466384661<A u 3 , A u 3 >_{-1}^0.5 =  1.2589535377386467e-09
Newton iteration   4energy = 8.680158466384663<A u 4 , A u 4 >_{-1}^0.5 =  9.90942431883319e-15
loadstep 29
Newton iteration   0energy = 8.676608400469208<A u 0 , A u 0 >_{-1}^0.5 =  0.005883036026966293
Newton iteration   1energy = 8.676591616023208<A u 1 , A u 1 >_{-1}^0.5 =  0.0006607892405581196
Newton iteration   2energy = 8.676591397476551<A u 2 , A u 2 >_{-1}^0.5 =  4.539372421135213e-06
Newton iteration   3energy = 8.676591397466245<A u 3 , A u 3 >_{-1}^0.5 =  8.403590576691722e-10
Newton iteration   4energy = 8.676591397466247<A u 4 , A u 4 >_{-1}^0.5 =  9.866077327214238e-15
loadstep 30
Newton iteration   0energy = 8.673007621183567<A u 0 , A u 0 >_{-1}^0.5 =  0.005730270629910569
Newton iteration   1energy = 8.67299166011491<A u 1 , A u 1 >_{-1}^0.5 =  0.0005989966873394284
Newton iteration   2energy = 8.67299148053809<A u 2 , A u 2 >_{-1}^0.5 =  3.7670865979156124e-06
Newton iteration   3energy = 8.672991480530989<A u 3 , A u 3 >_{-1}^0.5 =  5.648759494872811e-10
Newton iteration   4energy = 8.672991480530989<A u 4 , A u 4 >_{-1}^0.5 =  1.0050875893853173e-14
loadstep 31
Newton iteration   0energy = 8.66937569453993<A u 0 , A u 0 >_{-1}^0.5 =  0.0055862487812907875
Newton iteration   1energy = 8.669360494985764<A u 1 , A u 1 >_{-1}^0.5 =  0.0005444367972390969
Newton iteration   2energy = 8.66936034663853<A u 2 , A u 2 >_{-1}^0.5 =  3.136627189130437e-06
Newton iteration   3energy = 8.669360346633617<A u 3 , A u 3 >_{-1}^0.5 =  3.8233940124142637e-10
Newton iteration   4energy = 8.669360346633615<A u 4 , A u 4 >_{-1}^0.5 =  1.1734935299802074e-14
loadstep 32
Newton iteration   0energy = 8.665714114843603<A u 0 , A u 0 >_{-1}^0.5 =  0.005450309821622411
Newton iteration   1energy = 8.665699620344926<A u 1 , A u 1 >_{-1}^0.5 =  0.0004961330477172054
Newton iteration   2energy = 8.665699497158188<A u 2 , A u 2 >_{-1}^0.5 =  2.620341391709445e-06
Newton iteration   3energy = 8.665699497154765<A u 3 , A u 3 >_{-1}^0.5 =  2.6056570946754966e-10
Newton iteration   4energy = 8.665699497154764<A u 4 , A u 4 >_{-1}^0.5 =  1.217201318445093e-14
loadstep 33
Newton iteration   0energy = 8.662024260583653<A u 0 , A u 0 >_{-1}^0.5 =  0.005321851819893844
Newton iteration   1energy = 8.662010419633267<A u 1 , A u 1 >_{-1}^0.5 =  0.0004532560862173065
Newton iteration   2energy = 8.662010316822602<A u 2 , A u 2 >_{-1}^0.5 =  2.1962173093012577e-06
Newton iteration   3energy = 8.662010316820195<A u 3 , A u 3 >_{-1}^0.5 =  1.787715409258059e-10
Newton iteration   4energy = 8.662010316820183<A u 4 , A u 4 >_{-1}^0.5 =  1.067211648222395e-14
loadstep 34
Newton iteration   0energy = 8.658307405813543<A u 0 , A u 0 >_{-1}^0.5 =  0.005200325917091465
Newton iteration   1energy = 8.658294171419934<A u 1 , A u 1 >_{-1}^0.5 =  0.0004150994634587661
Newton iteration   2energy = 8.65829408519383<A u 2 , A u 2 >_{-1}^0.5 =  1.8466987982871333e-06
Newton iteration   3energy = 8.65829408519212<A u 3 , A u 3 >_{-1}^0.5 =  1.2346713844118924e-10
Newton iteration   4energy = 8.658294085192125<A u 4 , A u 4 >_{-1}^0.5 =  9.32755969194597e-15
loadstep 35
Newton iteration   0energy = 8.654564730219906<A u 0 , A u 0 >_{-1}^0.5 =  0.005085231193936819
Newton iteration   1energy = 8.654552059495636<A u 1 , A u 1 >_{-1}^0.5 =  0.00038105966948391123
Newton iteration   2energy = 8.654551986834138<A u 2 , A u 2 >_{-1}^0.5 =  1.557751764245773e-06
Newton iteration   3energy = 8.654551986832924<A u 3 , A u 3 >_{-1}^0.5 =  8.582341612129994e-11
Newton iteration   4energy = 8.654551986832924<A u 4 , A u 4 >_{-1}^0.5 =  1.2058191498589807e-14
loadstep 36
Newton iteration   0energy = 8.650797328048808<A u 0 , A u 0 >_{-1}^0.5 =  0.004976110032586554
Newton iteration   1energy = 8.65078518182817<A u 1 , A u 1 >_{-1}^0.5 =  0.00035061966629611104
Newton iteration   2energy = 8.650785120313985<A u 2 , A u 2 >_{-1}^0.5 =  1.3181296369092865e-06
Newton iteration   3energy = 8.650785120313117<A u 3 , A u 3 >_{-1}^0.5 =  6.003349278006359e-11
Newton iteration   4energy = 8.650785120313119<A u 4 , A u 4 >_{-1}^0.5 =  1.0325850670695882e-14
loadstep 37
Newton iteration   0energy = 8.647006216035676<A u 0 , A u 0 >_{-1}^0.5 =  0.004872543936238474
Newton iteration   1energy = 8.646994558523108<A u 1 , A u 1 >_{-1}^0.5 =  0.00032333526738206875
Newton iteration   2energy = 8.646994506212032<A u 2 , A u 2 >_{-1}^0.5 =  1.1187957572450256e-06
Newton iteration   3energy = 8.646994506211405<A u 3 , A u 3 >_{-1}^0.5 =  4.2252692019247613e-11
Newton iteration   4energy = 8.646994506211412<A u 4 , A u 4 >_{-1}^0.5 =  1.0729852040909942e-14
loadstep 38
Newton iteration   0energy = 8.64319234046425<A u 0 , A u 0 >_{-1}^0.5 =  0.004774149768405117
Newton iteration   1energy = 8.64318113891363<A u 1 , A u 1 >_{-1}^0.5 =  0.00029882384028847404
Newton iteration   2energy = 8.643181094234647<A u 2 , A u 2 >_{-1}^0.5 =  9.52469007074102e-07
Newton iteration   3energy = 8.643181094234198<A u 3 , A u 3 >_{-1}^0.5 =  2.99157638633773e-11
Newton iteration   4energy = 8.643181094234194<A u 4 , A u 4 >_{-1}^0.5 =  1.231791869358724e-14
loadstep 39
Newton iteration   0energy = 8.639356583463016<A u 0 , A u 0 >_{-1}^0.5 =  0.004680576373940036
Newton iteration   1energy = 8.639345807886512<A u 1 , A u 1 >_{-1}^0.5 =  0.0002767549095867613
Newton iteration   2energy = 8.639345769564418<A u 2 , A u 2 >_{-1}^0.5 =  8.132660428051996e-07
Newton iteration   3energy = 8.639345769564084<A u 3 , A u 3 >_{-1}^0.5 =  2.1305856722228868e-11
Newton iteration   4energy = 8.639345769564079<A u 4 , A u 4 >_{-1}^0.5 =  1.1818176358974371e-14
loadstep 40
Newton iteration   0energy = 8.635499768632432<A u 0 , A u 0 >_{-1}^0.5 =  0.004591501544822213
Newton iteration   1energy = 8.635489391537332<A u 1 , A u 1 >_{-1}^0.5 =  0.0002568423180182777
Newton iteration   2energy = 8.635489358532457<A u 2 , A u 2 >_{-1}^0.5 =  6.96419178009762e-07
Newton iteration   3energy = 8.635489358532212<A u 3 , A u 3 >_{-1}^0.5 =  1.5262192556902303e-11
Newton iteration   4energy = 8.635489358532213<A u 4 , A u 4 >_{-1}^0.5 =  1.1024939938001828e-14
loadstep 41
Newton iteration   0energy = 8.631622666083985<A u 0 , A u 0 >_{-1}^0.5 =  0.004506629296485266
Newton iteration   1energy = 8.631612662235378<A u 1 , A u 1 >_{-1}^0.5 =  0.0002388376690438732
Newton iteration   2energy = 8.631612633696458<A u 2 , A u 2 >_{-1}^0.5 =  5.980535273693957e-07
Newton iteration   3energy = 8.631612633696278<A u 3 , A u 3 >_{-1}^0.5 =  1.099135182257061e-11
Newton iteration   4energy = 8.631612633696287<A u 4 , A u 4 >_{-1}^0.5 =  1.1420765638026738e-14
loadstep 42
Newton iteration   0energy = 8.627725996961052<A u 0 , A u 0 >_{-1}^0.5 =  0.0044256874227469566
Newton iteration   1energy = 8.627716343168219<A u 1 , A u 1 >_{-1}^0.5 =  0.00022252482582335286
Newton iteration   2energy = 8.627716318395358<A u 2 , A u 2 >_{-1}^0.5 =  5.150106027527706e-07
Newton iteration   3energy = 8.627716318395228<A u 3 , A u 3 >_{-1}^0.5 =  7.959435264709924e-12
Newton iteration   4energy = 8.627716318395228<A u 4 , A u 4 >_{-1}^0.5 =  1.1578943244195082e-14
loadstep 43
Newton iteration   0energy = 8.62381043750256<A u 0 , A u 0 >_{-1}^0.5 =  0.0043484253001886426
Newton iteration   1energy = 8.62380111242687<A u 1 , A u 1 >_{-1}^0.5 =  0.00020771528351786156
Newton iteration   2energy = 8.623801090842262<A u 2 , A u 2 >_{-1}^0.5 =  4.447083520833389e-07
Newton iteration   3energy = 8.623801090842166<A u 3 , A u 3 >_{-1}^0.5 =  5.793345816132156e-12
Newton iteration   4energy = 8.623801090842163<A u 4 , A u 4 >_{-1}^0.5 =  1.0945054837429887e-14
loadstep 44
Newton iteration   0energy = 8.619876622702083<A u 0 , A u 0 >_{-1}^0.5 =  0.004274611915556383
Newton iteration   1energy = 8.619867606684437<A u 1 , A u 1 >_{-1}^0.5 =  0.0001942442655680472
Newton iteration   2energy = 8.619867587809216<A u 2 , A u 2 >_{-1}^0.5 =  3.850298436002284e-07
Newton iteration   3energy = 8.619867587809129<A u 3 , A u 3 >_{-1}^0.5 =  4.240958949701815e-12
Newton iteration   4energy = 8.619867587809136<A u 4 , A u 4 >_{-1}^0.5 =  1.1418242765222334e-14
loadstep 45
Newton iteration   0energy = 8.615925149608909<A u 0 , A u 0 >_{-1}^0.5 =  0.004204034092581162
Newton iteration   1energy = 8.615916424514648<A u 1 , A u 1 >_{-1}^0.5 =  0.00018196742192939653
Newton iteration   2energy = 8.615916407950385<A u 2 , A u 2 >_{-1}^0.5 =  3.3423451071494996e-07
Newton iteration   3energy = 8.61591640795033<A u 3 , A u 3 >_{-1}^0.5 =  3.1185096160350177e-12
Newton iteration   4energy = 8.615916407950332<A u 4 , A u 4 >_{-1}^0.5 =  1.110429334265253e-14
loadstep 46
Newton iteration   0energy = 8.611956580310858<A u 0 , A u 0 >_{-1}^0.5 =  0.004136494896624677
Newton iteration   1energy = 8.611948129390502<A u 1 , A u 1 >_{-1}^0.5 =  0.0001707580288323152
Newton iteration   2energy = 8.611948114804498<A u 2 , A u 2 >_{-1}^0.5 =  2.9088719186477544e-07
Newton iteration   3energy = 8.611948114804452<A u 3 , A u 3 >_{-1}^0.5 =  2.305393316626788e-12
Newton iteration   4energy = 8.611948114804456<A u 4 , A u 4 >_{-1}^0.5 =  1.2157063268037797e-14
loadstep 47
Newton iteration   0energy = 8.607971444634586<A u 0 , A u 0 >_{-1}^0.5 =  0.004071812198364899
Newton iteration   1energy = 8.607963252398571<A u 1 , A u 1 >_{-1}^0.5 =  0.0001605046079768612
Newton iteration   2energy = 8.607963239511943<A u 2 , A u 2 >_{-1}^0.5 =  2.5380125196407897e-07
Newton iteration   3energy = 8.60796323951191<A u 3 , A u 3 >_{-1}^0.5 =  1.710252507370915e-12
Newton iteration   4energy = 8.607963239511916<A u 4 , A u 4 >_{-1}^0.5 =  1.2290069453583734e-14
loadstep 48
Newton iteration   0energy = 8.603970242593698<A u 0 , A u 0 >_{-1}^0.5 =  0.0040098173792995325
Newton iteration   1energy = 8.603962294699814<A u 1 , A u 1 >_{-1}^0.5 =  0.00015110889702384532
Newton iteration   2energy = 8.60396228327801<A u 2 , A u 2 >_{-1}^0.5 =  2.219928560793886e-07
Newton iteration   3energy = 8.603962283277987<A u 3 , A u 3 >_{-1}^0.5 =  1.2781909165814276e-12
Newton iteration   4energy = 8.603962283277989<A u 4 , A u 4 >_{-1}^0.5 =  1.2774778602252051e-14
loadstep 49
Newton iteration   0energy = 8.599953446612274<A u 0 , A u 0 >_{-1}^0.5 =  0.003950354164065159
Newton iteration   1energy = 8.599945729764308<A u 1 , A u 1 >_{-1}^0.5 =  0.00014248411538807357
Newton iteration   2energy = 8.599945719609346<A u 2 , A u 2 >_{-1}^0.5 =  1.9464412525932455e-07
Newton iteration   3energy = 8.599945719609325<A u 3 , A u 3 >_{-1}^0.5 =  9.577059269412203e-13
Newton iteration   4energy = 8.599945719609325<A u 4 , A u 4 >_{-1}^0.5 =  1.1386571464267781e-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.

[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 = 0.4145019207496362<A u 0 , A u 0 >_{-1}^0.5 =  0.28092704638147636
Newton iteration   1energy = 0.37689785802772957<A u 1 , A u 1 >_{-1}^0.5 =  0.019174370471939796
Newton iteration   2energy = 0.3767134210404821<A u 2 , A u 2 >_{-1}^0.5 =  9.628937134596328e-05
Newton iteration   3energy = 0.3767134164045804<A u 3 , A u 3 >_{-1}^0.5 =  2.505456303796458e-09
Newton iteration   4energy = 0.3767134164045804<A u 4 , A u 4 >_{-1}^0.5 =  2.8167247591431503e-16
t =  0.1
Newton iteration   0energy = 0.34388128652385935<A u 0 , A u 0 >_{-1}^0.5 =  0.2360393474078292
Newton iteration   1energy = 0.31710944329874896<A u 1 , A u 1 >_{-1}^0.5 =  0.013442999343193951
Newton iteration   2energy = 0.31701887934635165<A u 2 , A u 2 >_{-1}^0.5 =  4.660092984274442e-05
Newton iteration   3energy = 0.31701887826051933<A u 3 , A u 3 >_{-1}^0.5 =  5.768048486573859e-10
Newton iteration   4energy = 0.3170188782605194<A u 4 , A u 4 >_{-1}^0.5 =  3.354660656138913e-16
t =  0.2
Newton iteration   0energy = 0.295494356513846<A u 0 , A u 0 >_{-1}^0.5 =  0.18345925931027088
Newton iteration   1energy = 0.2791640303389012<A u 1 , A u 1 >_{-1}^0.5 =  0.008014651163108922
Newton iteration   2energy = 0.27913186993623224<A u 2 , A u 2 >_{-1}^0.5 =  1.6250158077613474e-05
Newton iteration   3energy = 0.279131869804198<A u 3 , A u 3 >_{-1}^0.5 =  6.857117247134904e-11
Newton iteration   4energy = 0.279131869804198<A u 4 , A u 4 >_{-1}^0.5 =  3.7828954071306115e-16
t =  0.30000000000000004
Newton iteration   0energy = 0.26692789507920534<A u 0 , A u 0 >_{-1}^0.5 =  0.1334965893056886
Newton iteration   1energy = 0.25820503551107077<A u 1 , A u 1 >_{-1}^0.5 =  0.004188513106581715
Newton iteration   2energy = 0.258196257629024<A u 2 , A u 2 >_{-1}^0.5 =  4.3667568189871645e-06
Newton iteration   3energy = 0.2581962576194896<A u 3 , A u 3 >_{-1}^0.5 =  4.850620349308043e-12
Newton iteration   4energy = 0.25819625761948956<A u 4 , A u 4 >_{-1}^0.5 =  3.345785440454107e-16
t =  0.4
Newton iteration   0energy = 0.25204311423689185<A u 0 , A u 0 >_{-1}^0.5 =  0.09236175483868572
Newton iteration   1energy = 0.24783887867786306<A u 1 , A u 1 >_{-1}^0.5 =  0.0019843467589302334
Newton iteration   2energy = 0.2478369092241839<A u 2 , A u 2 >_{-1}^0.5 =  9.685217012146076e-07
Newton iteration   3energy = 0.24783690922371493<A u 3 , A u 3 >_{-1}^0.5 =  2.34741928405964e-13
Newton iteration   4energy = 0.24783690922371482<A u 4 , A u 4 >_{-1}^0.5 =  3.7019931635429417e-16
t =  0.5
Newton iteration   0energy = 0.24499322447001157<A u 0 , A u 0 >_{-1}^0.5 =  0.06164684190451959
Newton iteration   1energy = 0.24311100787346115<A u 1 , A u 1 >_{-1}^0.5 =  0.0008779623506489436
Newton iteration   2energy = 0.24311062240965625<A u 2 , A u 2 >_{-1}^0.5 =  1.8822078759980443e-07
Newton iteration   3energy = 0.24311062240963857<A u 3 , A u 3 >_{-1}^0.5 =  8.812738401378623e-15
t =  0.6
Newton iteration   0energy = 0.24187354814794007<A u 0 , A u 0 >_{-1}^0.5 =  0.0401663587565137
Newton iteration   1energy = 0.2410718102821203<A u 1 , A u 1 >_{-1}^0.5 =  0.000371536748024391
Newton iteration   2energy = 0.24107174125819195<A u 2 , A u 2 >_{-1}^0.5 =  3.366894131780417e-08
Newton iteration   3energy = 0.24107174125819136<A u 3 , A u 3 >_{-1}^0.5 =  5.177430126156407e-16
t =  0.7
Newton iteration   0energy = 0.24055438873764573<A u 0 , A u 0 >_{-1}^0.5 =  0.02578203385924664
Newton iteration   1energy = 0.2402233339155409<A u 1 , A u 1 >_{-1}^0.5 =  0.00015331214259509088
Newton iteration   2energy = 0.2402233221629405<A u 2 , A u 2 >_{-1}^0.5 =  5.795968720638723e-09
Newton iteration   3energy = 0.24022332216294048<A u 3 , A u 3 >_{-1}^0.5 =  4.1383921569936535e-16
t =  0.7999999999999999
Newton iteration   0energy = 0.2400118618581594<A u 0 , A u 0 >_{-1}^0.5 =  0.016425332383409572
Newton iteration   1energy = 0.23987730469401894<A u 1 , A u 1 >_{-1}^0.5 =  6.283809116183802e-05
Newton iteration   2energy = 0.23987730271968508<A u 2 , A u 2 >_{-1}^0.5 =  1.0154930579076878e-09
Newton iteration   3energy = 0.23987730271968513<A u 3 , A u 3 >_{-1}^0.5 =  4.6394360215478465e-16
t =  0.8999999999999999
Newton iteration   0energy = 0.2397915819363539<A u 0 , A u 0 >_{-1}^0.5 =  0.010464445156609927
Newton iteration   1energy = 0.2397369188905994<A u 1 , A u 1 >_{-1}^0.5 =  2.620620881243329e-05
Newton iteration   2energy = 0.23973691854721507<A u 2 , A u 2 >_{-1}^0.5 =  1.9994608625477876e-10
Newton iteration   3energy = 0.23973691854721502<A u 3 , A u 3 >_{-1}^0.5 =  4.712449072357031e-16
t =  0.9999999999999999
Newton iteration   0energy = 0.23970186094053136<A u 0 , A u 0 >_{-1}^0.5 =  0.006735415826965045
Newton iteration   1energy = 0.2396792028920524<A u 1 , A u 1 >_{-1}^0.5 =  1.1556763084997694e-05
Newton iteration   2energy = 0.23967920282527277<A u 2 , A u 2 >_{-1}^0.5 =  5.099460594642426e-11
Newton iteration   3energy = 0.2396792028252729<A u 3 , A u 3 >_{-1}^0.5 =  4.3603056960577796e-16
t =  1.0999999999999999
Newton iteration   0energy = 0.23966435830447483<A u 0 , A u 0 >_{-1}^0.5 =  0.004452542361403728
Newton iteration   1energy = 0.23965445350959919<A u 1 , A u 1 >_{-1}^0.5 =  5.694142646688917e-06
Newton iteration   2energy = 0.23965445349338754<A u 2 , A u 2 >_{-1}^0.5 =  1.772503968582708e-11
Newton iteration   3energy = 0.2396544534933875<A u 3 , A u 3 >_{-1}^0.5 =  4.506057912655227e-16
t =  1.2
Newton iteration   0energy = 0.23964767142737936<A u 0 , A u 0 >_{-1}^0.5 =  0.003097908141817561
Newton iteration   1energy = 0.23964287580252738<A u 1 , A u 1 >_{-1}^0.5 =  3.2840536759779764e-06
Newton iteration   2energy = 0.23964287579713486<A u 2 , A u 2 >_{-1}^0.5 =  7.767219511989633e-12
Newton iteration   3energy = 0.2396428757971348<A u 3 , A u 3 >_{-1}^0.5 =  4.61045832186731e-16
t =  1.3
Newton iteration   0energy = 0.2396393498836377<A u 0 , A u 0 >_{-1}^0.5 =  0.002326255780763519
Newton iteration   1energy = 0.23963664545354452<A u 1 , A u 1 >_{-1}^0.5 =  2.2181767038734202e-06
Newton iteration   2energy = 0.23963664545108435<A u 2 , A u 2 >_{-1}^0.5 =  3.983596215905177e-12
Newton iteration   3energy = 0.23963664545108426<A u 3 , A u 3 >_{-1}^0.5 =  4.282835092824085e-16
t =  1.4000000000000001
Newton iteration   0energy = 0.2396344792626695<A u 0 , A u 0 >_{-1}^0.5 =  0.0019006816819442077
Newton iteration   1energy = 0.2396326736515232<A u 1 , A u 1 >_{-1}^0.5 =  1.687946260850493e-06
Newton iteration   2energy = 0.23963267365009858<A u 2 , A u 2 >_{-1}^0.5 =  2.2982453014628886e-12
Newton iteration   3energy = 0.23963267365009855<A u 3 , A u 3 >_{-1}^0.5 =  4.3568724144101285e-16
t =  1.5000000000000002
Newton iteration   0energy = 0.2396311154900338<A u 0 , A u 0 >_{-1}^0.5 =  0.0016629280504841238
Newton iteration   1energy = 0.23962973322336592<A u 1 , A u 1 >_{-1}^0.5 =  1.3855651852122846e-06
Newton iteration   2energy = 0.2396297332224059<A u 2 , A u 2 >_{-1}^0.5 =  1.4586272866410796e-12
Newton iteration   3energy = 0.23962973322240597<A u 3 , A u 3 >_{-1}^0.5 =  4.4042630649306477e-16
t =  1.6000000000000003
Newton iteration   0energy = 0.23962848049094232<A u 0 , A u 0 >_{-1}^0.5 =  0.0015187632040499007
Newton iteration   1energy = 0.2396273274180997<A u 1 , A u 1 >_{-1}^0.5 =  1.1887244241389382e-06
Newton iteration   2energy = 0.23962732741739323<A u 2 , A u 2 >_{-1}^0.5 =  1.0010925224110844e-12
Newton iteration   3energy = 0.23962732741739318<A u 3 , A u 3 >_{-1}^0.5 =  4.4047895924105704e-16
t =  1.7000000000000004
Newton iteration   0energy = 0.2396262539867423<A u 0 , A u 0 >_{-1}^0.5 =  0.0014193843873494107
Newton iteration   1energy = 0.23962524682330955<A u 1 , A u 1 >_{-1}^0.5 =  1.045632909422474e-06
Newton iteration   2energy = 0.2396252468227629<A u 2 , A u 2 >_{-1}^0.5 =  7.300861400170568e-13
Newton iteration   3energy = 0.23962524682276293<A u 3 , A u 3 >_{-1}^0.5 =  4.513659338716933e-16
t =  1.8000000000000005
Newton iteration   0energy = 0.23962429644294805<A u 0 , A u 0 >_{-1}^0.5 =  0.0013419591604952733
Newton iteration   1energy = 0.2396233961271811<A u 1 , A u 1 >_{-1}^0.5 =  9.33053216472548e-07
Newton iteration   2energy = 0.2396233961267457<A u 2 , A u 2 >_{-1}^0.5 =  5.563795108964307e-13
Newton iteration   3energy = 0.23962339612674574<A u 3 , A u 3 >_{-1}^0.5 =  4.0552927438094465e-16
t =  1.9000000000000006
Newton iteration   0energy = 0.23962254072403263<A u 0 , A u 0 >_{-1}^0.5 =  0.0012763226584933272
Newton iteration   1energy = 0.2396217263037931<A u 1 , A u 1 >_{-1}^0.5 =  8.399033658894825e-07
Newton iteration   2energy = 0.23962172630344045<A u 2 , A u 2 >_{-1}^0.5 =  4.3696359967523316e-13
Newton iteration   3energy = 0.23962172630344042<A u 3 , A u 3 >_{-1}^0.5 =  4.279057390248914e-16
t =  2.0000000000000004
Newton iteration   0energy = 0.2396209495967649<A u 0 , A u 0 >_{-1}^0.5 =  0.0012179715961899576
Newton iteration   1energy = 0.23962020792824323<A u 1 , A u 1 >_{-1}^0.5 =  7.604668960240027e-07
Newton iteration   2energy = 0.23962020792795402<A u 2 , A u 2 >_{-1}^0.5 =  3.502994241571864e-13
Newton iteration   3energy = 0.2396202079279541<A u 3 , A u 3 >_{-1}^0.5 =  4.1540976898622703e-16
t =  2.1000000000000005
Newton iteration   0energy = 0.2396194989264497<A u 0 , A u 0 >_{-1}^0.5 =  0.0011648358763689138
Newton iteration   1energy = 0.23961882055016812<A u 1 , A u 1 >_{-1}^0.5 =  6.914942047541799e-07
Newton iteration   2energy = 0.23961882054992892<A u 2 , A u 2 >_{-1}^0.5 =  2.849344039398136e-13
Newton iteration   3energy = 0.23961882054992895<A u 3 , A u 3 >_{-1}^0.5 =  4.022060791912109e-16
t =  2.2000000000000006
Newton iteration   0energy = 0.2396181709174179<A u 0 , A u 0 >_{-1}^0.5 =  0.001115880689979333
Newton iteration   1energy = 0.23961754835794208<A u 1 , A u 1 >_{-1}^0.5 =  6.309368258332487e-07
Newton iteration   2energy = 0.23961754835774315<A u 2 , A u 2 >_{-1}^0.5 =  2.342351024452693e-13
Newton iteration   3energy = 0.2396175483577431<A u 3 , A u 3 >_{-1}^0.5 =  4.124775316997188e-16
t =  2.3000000000000007
Newton iteration   0energy = 0.23961695130021216<A u 0 , A u 0 >_{-1}^0.5 =  0.0010705168654718392
Newton iteration   1energy = 0.2396163783254567<A u 1 , A u 1 >_{-1}^0.5 =  5.773776868941566e-07
Newton iteration   2energy = 0.23961637832529006<A u 2 , A u 2 >_{-1}^0.5 =  1.9415448405273936e-13
Newton iteration   3energy = 0.23961637832529<A u 3 , A u 3 >_{-1}^0.5 =  4.0997221812292823e-16
t =  2.400000000000001
Newton iteration   0energy = 0.23961582808585116<A u 0 , A u 0 >_{-1}^0.5 =  0.0010283540627855545
Newton iteration   1energy = 0.23961529935305362<A u 1 , A u 1 >_{-1}^0.5 =  5.297627567863049e-07
Newton iteration   2energy = 0.23961529935291334<A u 2 , A u 2 >_{-1}^0.5 =  1.6199821162823026e-13
Newton iteration   3energy = 0.23961529935291334<A u 3 , A u 3 >_{-1}^0.5 =  4.0039485505058225e-16
t =  2.500000000000001
Newton iteration   0energy = 0.2396147909550374<A u 0 , A u 0 >_{-1}^0.5 =  0.0009890970729080236
Newton iteration   1energy = 0.2396143018177979<A u 1 , A u 1 >_{-1}^0.5 =  4.872666035318775e-07
Newton iteration   2energy = 0.2396143018176792<A u 2 , A u 2 >_{-1}^0.5 =  1.3598853043606795e-13
Newton iteration   3energy = 0.2396143018176792<A u 3 , A u 3 >_{-1}^0.5 =  4.323276194821206e-16
t =  2.600000000000001
Newton iteration   0energy = 0.23961383091454508<A u 0 , A u 0 >_{-1}^0.5 =  0.0009525015914657844
Newton iteration   1energy = 0.23961337730105753<A u 1 , A u 1 >_{-1}^0.5 =  4.4921981868305184e-07
Newton iteration   2energy = 0.23961337730095666<A u 2 , A u 2 >_{-1}^0.5 =  1.1473288840449472e-13
Newton iteration   3energy = 0.2396133773009567<A u 3 , A u 3 >_{-1}^0.5 =  4.1675889114543924e-16
t =  2.700000000000001
Newton iteration   0energy = 0.2396129400737677<A u 0 , A u 0 >_{-1}^0.5 =  0.0009183547757921798
Newton iteration   1energy = 0.23961251839967956<A u 1 , A u 1 >_{-1}^0.5 =  4.150665657909605e-07
Newton iteration   2energy = 0.2396125183995935<A u 2 , A u 2 >_{-1}^0.5 =  9.724524553808296e-14
t =  2.800000000000001
Newton iteration   0energy = 0.2396121114812304<A u 0 , A u 0 >_{-1}^0.5 =  0.0008864662194052857
Newton iteration   1energy = 0.23961171858168248<A u 1 , A u 1 >_{-1}^0.5 =  3.843377621229476e-07
Newton iteration   2energy = 0.23961171858160873<A u 2 , A u 2 >_{-1}^0.5 =  8.279056742710236e-14
t =  2.9000000000000012
Newton iteration   0energy = 0.2396113389953494<A u 0 , A u 0 >_{-1}^0.5 =  0.0008566633749443014
Newton iteration   1energy = 0.2396109720692427<A u 1 , A u 1 >_{-1}^0.5 =  3.5663295477607706e-07
Newton iteration   2energy = 0.23961097206917908<A u 2 , A u 2 >_{-1}^0.5 =  7.079233340119339e-14
t =  3.0000000000000013
Newton iteration   0energy = 0.2396106171776509<A u 0 , A u 0 >_{-1}^0.5 =  0.0008287889263646606
Newton iteration   1energy = 0.23961027374068344<A u 1 , A u 1 >_{-1}^0.5 =  3.316073784868388e-07
Newton iteration   2energy = 0.2396102737406285<A u 2 , A u 2 >_{-1}^0.5 =  6.084230609935835e-14
t =  3.1000000000000014
Newton iteration   0energy = 0.23960994120243975<A u 0 , A u 0 >_{-1}^0.5 =  0.000802699050678995
Newton iteration   1energy = 0.23960961904696895<A u 1 , A u 1 >_{-1}^0.5 =  3.0896230208916104e-07
Newton iteration   2energy = 0.2396096190469213<A u 2 , A u 2 >_{-1}^0.5 =  5.2419355816600374e-14
t =  3.2000000000000015
Newton iteration   0energy = 0.23960930677943418<A u 0 , A u 0 >_{-1}^0.5 =  0.0007782621114844118
Newton iteration   1energy = 0.23960900393990864<A u 1 , A u 1 >_{-1}^0.5 =  2.8843758042950464e-07
Newton iteration   2energy = 0.239609003939867<A u 2 , A u 2 >_{-1}^0.5 =  4.53837460691867e-14
t =  3.3000000000000016
Newton iteration   0energy = 0.23960871008705956<A u 0 , A u 0 >_{-1}^0.5 =  0.0007553575814402505
Newton iteration   1energy = 0.2396084248101213<A u 1 , A u 1 >_{-1}^0.5 =  2.6980575459079707e-07
Newton iteration   2energy = 0.23960842481008487<A u 2 , A u 2 >_{-1}^0.5 =  3.943490918739899e-14
t =  3.4000000000000017
Newton iteration   0energy = 0.239608147714714<A u 0 , A u 0 >_{-1}^0.5 =  0.0007338751008235469
Newton iteration   1energy = 0.23960787843327308<A u 1 , A u 1 >_{-1}^0.5 =  2.528672818809477e-07
Newton iteration   2energy = 0.23960787843324113<A u 2 , A u 2 >_{-1}^0.5 =  3.4395575841844475e-14
t =  3.5000000000000018
Newton iteration   0energy = 0.2396076166126835<A u 0 , A u 0 >_{-1}^0.5 =  0.0007137136279143667
Newton iteration   1energy = 0.23960736192339643<A u 1 , A u 1 >_{-1}^0.5 =  2.3744661572988932e-07
Newton iteration   2energy = 0.23960736192336823<A u 2 , A u 2 >_{-1}^0.5 =  3.010172589440937e-14
t =  3.600000000000002
Newton iteration   0energy = 0.23960711404861693<A u 0 , A u 0 >_{-1}^0.5 =  0.0006947806591329083
Newton iteration   1energy = 0.23960687269229805<A u 1 , A u 1 >_{-1}^0.5 =  2.2338894231226247e-07
Newton iteration   2energy = 0.23960687269227302<A u 2 , A u 2 >_{-1}^0.5 =  2.6431184772178026e-14
t =  3.700000000000002
Newton iteration   0energy = 0.23960663756966022<A u 0 , A u 0 >_{-1}^0.5 =  0.0006769915073033805
Newton iteration   1energy = 0.2396064084142239<A u 1 , A u 1 >_{-1}^0.5 =  2.1055743624060828e-07
Newton iteration   2energy = 0.2396064084142018<A u 2 , A u 2 >_{-1}^0.5 =  2.3344047294241317e-14
t =  3.800000000000002
Newton iteration   0energy = 0.23960618496947494<A u 0 , A u 0 >_{-1}^0.5 =  0.0006602686315022897
Newton iteration   1energy = 0.23960596699506795<A u 1 , A u 1 >_{-1}^0.5 =  1.988309330958618e-07
Newton iteration   2energy = 0.2396059669950483<A u 2 , A u 2 >_{-1}^0.5 =  2.0667323036294112e-14
t =  3.900000000000002
Newton iteration   0energy = 0.23960575425948313<A u 0 , A u 0 >_{-1}^0.5 =  0.0006445410144724198
Newton iteration   1energy = 0.23960554654551278<A u 1 , A u 1 >_{-1}^0.5 =  1.881019423294816e-07
Newton iteration   2energy = 0.2396055465454951<A u 2 , A u 2 >_{-1}^0.5 =  1.8392439447062177e-14
t =  4.000000000000002
Newton iteration   0energy = 0.239605343643775<A u 0 , A u 0 >_{-1}^0.5 =  0.0006297435848349288
Newton iteration   1energy = 0.2396051453575803<A u 1 , A u 1 >_{-1}^0.5 =  1.7827494164571733e-07
Newton iteration   2energy = 0.23960514535756447<A u 2 , A u 2 >_{-1}^0.5 =  1.6359534775821174e-14
t =  4.100000000000001
Newton iteration   0energy = 0.23960495149718894<A u 0 , A u 0 >_{-1}^0.5 =  0.000615816681932275
Newton iteration   1energy = 0.2396047618841374<A u 1 , A u 1 >_{-1}^0.5 =  1.6926490629353927e-07
Newton iteration   2energy = 0.23960476188412316<A u 2 , A u 2 >_{-1}^0.5 =  1.4650955748249466e-14
t =  4.200000000000001
Newton iteration   0energy = 0.23960457634614438<A u 0 , A u 0 >_{-1}^0.5 =  0.000602705561393757
Newton iteration   1energy = 0.23960439472096595<A u 1 , A u 1 >_{-1}^0.5 =  1.6099603651134076e-07
Newton iteration   2energy = 0.23960439472095302<A u 2 , A u 2 >_{-1}^0.5 =  1.316632014806186e-14
t =  4.300000000000001
Newton iteration   0energy = 0.23960421685186575<A u 0 , A u 0 >_{-1}^0.5 =  0.0005903599395996324
Newton iteration   1energy = 0.23960404259105958<A u 1 , A u 1 >_{-1}^0.5 =  1.534006526259395e-07
Newton iteration   2energy = 0.2396040425910478<A u 2 , A u 2 >_{-1}^0.5 =  1.1891481034967456e-14
t =  4.4
Newton iteration   0energy = 0.2396038717956791<A u 0 , A u 0 >_{-1}^0.5 =  0.0005787335752184119
Newton iteration   1energy = 0.23960370433085504<A u 1 , A u 1 >_{-1}^0.5 =  1.4641823370105469e-07
Newton iteration   2energy = 0.23960370433084427<A u 2 , A u 2 >_{-1}^0.5 =  1.0716509618600795e-14
t =  4.5
Newton iteration   0energy = 0.23960354006611384<A u 0 , A u 0 >_{-1}^0.5 =  0.0005677838859552442
Newton iteration   1energy = 0.23960337887814379<A u 1 , A u 1 >_{-1}^0.5 =  1.3999457888852992e-07
Newton iteration   2energy = 0.23960337887813393<A u 2 , A u 2 >_{-1}^0.5 =  9.745816221516653e-15
t =  4.6
Newton iteration   0energy = 0.2396032206475711<A u 0 , A u 0 >_{-1}^0.5 =  0.0005574715986072078
Newton iteration   1energy = 0.23960306526144695<A u 1 , A u 1 >_{-1}^0.5 =  1.3408107469814374e-07
Newton iteration   2energy = 0.2396030652614379<A u 2 , A u 2 >_{-1}^0.5 =  8.896636967396773e-15
t =  4.699999999999999
Newton iteration   0energy = 0.23960291261035707<A u 0 , A u 0 >_{-1}^0.5 =  0.0005477604304859979
Newton iteration   1energy = 0.23960276259066243<A u 1 , A u 1 >_{-1}^0.5 =  1.2863405339167213e-07
Newton iteration   2energy = 0.23960276259065416<A u 2 , A u 2 >_{-1}^0.5 =  8.158359737324676e-15
t =  4.799999999999999
Newton iteration   0energy = 0.23960261510190303<A u 0 , A u 0 >_{-1}^0.5 =  0.0005386168002506355
Newton iteration   1energy = 0.2396024700488202<A u 1 , A u 1 >_{-1}^0.5 =  1.2361423087280676e-07
Newton iteration   2energy = 0.2396024700488126<A u 2 , A u 2 >_{-1}^0.5 =  7.475013030957406e-15
t =  4.899999999999999
Newton iteration   0energy = 0.23960232733902043<A u 0 , A u 0 >_{-1}^0.5 =  0.0005300095661919717
Newton iteration   1energy = 0.2396021868848042<A u 1 , A u 1 >_{-1}^0.5 =  1.1898621270787481e-07
Newton iteration   2energy = 0.2396021868847971<A u 2 , A u 2 >_{-1}^0.5 =  6.9022263020530444e-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 =  0.39392955469293095
Newton iteration  1
err =  1.7898870223932246e-15

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

\[\int_{\Omega} \nabla u \cdot \nabla v = \int_{\Omega} 1 \cdot v \quad \forall ~ v \in V^{ho,disc}\]

under the constraint

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

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