This page was generated from unit-6.2-contact/contact.ipynb.

6.2 Contact Problems

work in progress …

from netgen.occ import *
from ngsolve import *
from ngsolve.solvers import *
from ngsolve.webgui import Draw
from netgen.webgui import Draw as DrawGeo

rect = MoveTo(0,-0.02).Rectangle(0.04, 0.02).Face()
ball = MoveTo(0,0).Arc(0.005, 90).Rotate(90).Line(0.005).Close().Face()

rect.edges.Min(X).name="sym"
rect.edges.Max(X).name="free"
rect.edges.Min(Y).name="fix"
rect.edges.Max(Y).name="contact1"
ball.edges.Min(X).name="sym"
ball.edges.Max(Y).name="disp"
ball.edges.Max(X-Y).name="contact2"
ball.faces.name="ball"
rect.faces.name="rect"
rect = Glue ([rect, Vertex((0.001,0,0))])
rect.vertices.Max(Y-X).maxh=0.0002
rect.edges.Max(Y-0.1*X).maxh=0.00002
geo = Compound([rect,ball])
DrawGeo (geo)
mesh = Mesh(OCCGeometry(geo, dim=2).GenerateMesh(maxh=0.002))
mesh.Curve(4)
Draw(mesh);

mesh.GetBoundaries(), mesh.GetMaterials()

(('fix', 'free', 'contact1', 'contact1', 'sym', 'contact2', 'disp', 'sym'),
 ('rect', 'ball'))

E = 2.1e11
nu = 0.2

rho = 7e3
mu  = E / 2 / (1+nu)
lam = E * nu / ((1+nu)*(1-2*nu))

I = Id(mesh.dim)
def Pow(a, b):
    return a**b  # exp (log(a)*b)

def C(u):
    # F = Id(2) + Grad(u)
    G = Grad(u)
    F = Id(3) + CoefficientFunction( (G[0,0], G[1,0], 0,   G[1,0], G[1,1], 0,    0, 0, u[0]/(x+1e-16) ) )
    return F.trans*F

def NeoHooke (C):
    return 0.5 * mu * (Trace(C-Id(C.dims[0])) + 2*mu/lam * Det(C)**(-lam/2/mu) - 1)

fes = VectorH1(mesh, order=4, dirichletx="sym", dirichlety="fix|disp")
u,v = fes.TnT()

a = BilinearForm(fes)
a += Variation((x*NeoHooke(C(u))).Compile()*dx)

contact = ContactBoundary(mesh.Boundaries("contact1"), mesh.Boundaries("contact2"))

X = CoefficientFunction((x,y))
cf = (X + u - (X.Other() + u.Other())) * contact.normal

# energy formulation
# contact.AddEnergy(IfPos(cf, 1e14*cf*cf, 0))
# or integrator
contact.AddIntegrator(IfPos(cf, 1e14*cf*(v-v.Other())*contact.normal, 0))

gfu = GridFunction(fes)

disp = -10e-6
gfu.Set((0, disp), definedon=mesh.Materials("ball"))

sceneu = Draw(gfu, deformation=gfu)

C_ = C(gfu).MakeVariable()
sigma = NeoHooke(C_).Diff(C_)
scenesigma = Draw(sigma[1,1], mesh, "sigmazz", min=-2e9, max=2e9)


with TaskManager():
    contact.Update(gfu, a, 200, 1e-4)

    NewtonMinimization(a=a, u=gfu, printing=False, inverse="sparsecholesky")
    sceneu.Redraw()
    scenesigma.Redraw()

Draw(sigma[1,1], mesh, "sigmazz", min=-5e9, max=0, deformation=gfu);

Dynamic Contact

from netgen.occ import *
from ngsolve import *
from ngsolve.solvers import *
from ngsolve.webgui import Draw
from netgen.webgui import Draw as DrawGeo

bowl = MoveTo(0,-0.01).Arc(0.2,70).Rotate(90).Line(0.01).Rotate(90) \
    .Arc(0.19,-140).Rotate(90).Line(0.01).Rotate(90).Arc(0.2,70).Face()
ball1 = Circle((0,0.1),0.01).Face()
ball2 = Circle((0.05,0.1),0.01).Face()
ball3 = Circle((-0.05,0.1),0.01).Face()
balls = Compound([ball1, ball2, ball3])
balls.edges.name = "balls"
geo = Compound([bowl, balls])
bowl.edges.name="contact"
bowl.edges.Min(Y+0.01*X).name="fix"
bowl.edges.Min(Y-0.01*X).name="fix"
DrawGeo(geo);

mesh = Mesh(OCCGeometry(geo, dim=2).GenerateMesh(maxh=0.005))
mesh.Curve(4)
Draw(mesh);

mesh.GetBoundaries()

('fix', 'contact', 'contact', 'contact', 'fix', 'balls', 'balls', 'balls')

E, nu = 210e6, 0.2
rho = 7e3
mu  = E / 2 / (1+nu)
lam = E * nu / ((1+nu)*(1-2*nu))

I = Id(mesh.dim)

def C(u):
    F = I+Grad(u)
    return F.trans*F
def NeoHooke (C):
    return 0.5 * mu * (Trace(C-I) + 2*mu/lam * Det(C)**(-lam/2/mu) - 1)

fes = VectorH1(mesh, order=3, dirichlet="fix")
u,v = fes.TnT()

force = CF((0,-9.81*rho))

uold = GridFunction(fes)
unew = GridFunction(fes)
vel = GridFunction(fes)
anew = GridFunction(fes)
aold = GridFunction(fes)

tau = 2e-4

bfmstar = BilinearForm(fes)
bfmstar += Variation( NeoHooke (C(u)).Compile(False)*dx )
bfmstar += Variation( -force*u*dx )
bfmstar += Variation( rho/2* 2/tau**2 * (u-uold-tau*vel-tau**2/4*aold)**2 * dx )

tend = 1

scene = Draw (unew, mesh, "disp", deformation=unew)

t = 0
unew.Set( (0,0) )
vel.Set( (0,0) )
contact = ContactBoundary(mesh.Boundaries("contact|balls"), mesh.Boundaries("contact|balls"))

X = CoefficientFunction((x,y))

if True:
    cf = (X + u-uold - (X.Other() + u.Other() - uold.Other())) * (-specialcf.normal(2).Other())
    # cf = (X + u-uold)*specialcf.normal(2) + \
    #    (X.Other() + u.Other() - uold.Other()) * specialcf.normal(2).Other()
    contact.AddEnergy(IfPos(cf, 1e9*cf*cf, 0), deformed=True)
else:
    cf = (X + u - (X.Other() + u.Other())) * contact.normal
    contact.AddEnergy(IfPos(cf, 1e9*cf*cf, 0), deformed=False)

with TaskManager():
    while t < tend:
        print ("time", t)
        t += tau
        uold.vec.data = unew.vec
        aold.vec.data = anew.vec

        contact.Update(uold, bfmstar, 5, 0.01)
        NewtonMinimization (a=bfmstar, u=unew, printing=False, inverse="sparsecholesky")

        anew.vec.data = unew.vec-uold.vec-tau*vel.vec-tau**2/4*aold.vec
        anew.vec.data *= 4/tau**2
        vel.vec.data += 0.5*tau*aold.vec
        vel.vec.data += 0.5*tau*anew.vec

        scene.Redraw()

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A 3D example

from netgen.occ import *
from ngsolve import *
from ngsolve.solvers import *
from ngsolve.webgui import Draw
from netgen.webgui import Draw as DrawGeo

box = Box((0,0,0), (1,1,0.5))
rod = Cylinder((0.5, 0.5, -0.5),Z,0.2,1.5)
rod.faces.name="contact1"
block = box-rod
block.faces.Min(X).name="fix"
rod.faces.name="contact2"
rod.faces.Max(Z).name="force"
rod.faces.Min(Z).name="free"
geo = Compound([block, rod])
DrawGeo (geo);
mesh = Mesh(OCCGeometry(geo).GenerateMesh(maxh=0.1)).Curve(3)
Draw(mesh);

mesh.GetBoundaries()

('fix',
 'default',
 'default',
 'default',
 'default',
 'default',
 'contact1',
 'contact2',
 'force',
 'free')

fes = VectorH1(mesh, order=3, dirichlet="fix")
u = fes.TrialFunction()
gfu = GridFunction(fes)

a = BilinearForm(fes)
a += Variation(InnerProduct(Sym(Grad(u)), Sym(Grad(u)))*dx)
a += Variation(1e-8*u*u*dx)   # regularization
a += Variation(-0.01*u[0]*ds("force"))

contact = ContactBoundary(mesh.Boundaries("contact1"), mesh.Boundaries("contact2"))
X = CoefficientFunction((x,y,z))
cf = (X + u - (X.Other() + u.Other())) * specialcf.normal(3)
contact.AddEnergy(IfPos(cf, 1e4*cf*cf, 0))
contact.Update(gfu, a, 20, 4)

with TaskManager(pajetrace=10**9):
    NewtonMinimization(a=a, u=gfu, printing=True, inverse="sparsecholesky");

Newton iteration  0
Energy:  1.878925160437614e-06
err =  0.01043671856331282
Newton iteration  1
Energy:  -5.18465761437002e-05
err =  0.0014588684464326566
Newton iteration  2
Energy:  -5.303290411662016e-05
err =  0.001205339816872748
Newton iteration  3
Energy:  -5.3887413267997095e-05
err =  0.001574259710384505
Newton iteration  4
Energy:  -5.524579405251159e-05
err =  0.0019497532008952627
Newton iteration  5
Energy:  -5.7350679492468387e-05
err =  0.0016222630623349487
Newton iteration  6
Energy:  -5.885787870527439e-05
err =  0.0015937895146848054
Newton iteration  7
Energy:  -6.037555150634093e-05
err =  0.0037870886525346796
Newton iteration  8
Energy:  -6.741305557718126e-05
err =  0.002134048040868824
Newton iteration  9
Energy:  -6.974669887738788e-05
err =  0.0011800061241293988
Newton iteration  10
Energy:  -7.052750774659336e-05
err =  0.0013080641009602454
Newton iteration  11
Energy:  -7.144383787382658e-05
err =  0.0008705401982222964
Newton iteration  12
Energy:  -7.196848923958954e-05
err =  0.0009966532221694674
Newton iteration  13
Energy:  -7.248329449043798e-05
err =  0.0004383531990137823
Newton iteration  14
Energy:  -7.256765890845204e-05
err =  0.000539322838758066
Newton iteration  15
Energy:  -7.271894180725873e-05
err =  0.0003254726227566483
Newton iteration  16
Energy:  -7.278242336879881e-05
err =  0.0002748157872067239
Newton iteration  17
Energy:  -7.281641605472725e-05
err =  0.0002555090887248473
Newton iteration  18
Energy:  -7.285228754353191e-05
err =  0.00019857823479227276
Newton iteration  19
Energy:  -7.287379089302546e-05
err =  0.0001566393419826139
Newton iteration  20
Energy:  -7.288609582197894e-05
err =  0.00011754091025303765
Newton iteration  21
Energy:  -7.289388544332841e-05
err =  8.084639575143286e-05
Newton iteration  22
Energy:  -7.289698223169251e-05
err =  6.473898980968857e-05
Newton iteration  23
Energy:  -7.289944854601201e-05
err =  6.017497276335911e-05
Newton iteration  24
Energy:  -7.290138165099542e-05
err =  1.9972753801502722e-05
Newton iteration  25
Energy:  -7.290160329561195e-05
err =  8.995833149198599e-06
Newton iteration  26
Energy:  -7.29016496288968e-05
err =  4.988705780312684e-06
Newton iteration  27
Energy:  -7.290166323571163e-05
err =  2.7617254219267412e-06
Newton iteration  28
Energy:  -7.290166715324943e-05
err =  4.904055438147451e-07
Newton iteration  29
Energy:  -7.290166727349816e-05
err =  9.686701544569373e-16

Draw (gfu, mesh, deformation=True);
Draw (Sym(Grad(gfu))[0,0], mesh, draw_surf=False);