# 6.2 Contact Problems¶

work in progress …

[1]:

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);

[2]:

mesh.GetBoundaries(), mesh.GetMaterials()

[2]:

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

[3]:

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)
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
# or integrator

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)

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¶

[4]:

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

[5]:

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);

[6]:

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

[7]:

mesh.GetBoundaries()

[7]:

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

[8]:

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):
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()
else:
cf = (X + u - (X.Other() + u.Other())) * contact.normal

[9]:

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¶

[10]:

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

[11]:

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);

[12]:

mesh.GetBoundaries()

[12]:

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

[13]:

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

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

[14]:

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

[15]:

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

Newton iteration  0
Energy:  1.878925160437614e-06
err =  0.010438120592163605
Newton iteration  1
Energy:  -5.186328141928307e-05
err =  0.0014552601622141747
Newton iteration  2
Energy:  -5.304357958552533e-05
err =  0.0012065619791319542
Newton iteration  3
Energy:  -5.390073896803565e-05
err =  0.0015795721271271235
Newton iteration  4
Energy:  -5.5269334824312883e-05
err =  0.0019466649148135687
Newton iteration  5
Energy:  -5.737007531066267e-05
err =  0.0016021188507444645
Newton iteration  6
Energy:  -5.875591978167442e-05
err =  0.0018120341568668976
Newton iteration  7
Energy:  -6.060015430167184e-05
err =  0.003844076476287025
Newton iteration  8
Energy:  -6.763929993935263e-05
err =  0.001937397803620593
Newton iteration  9
Energy:  -6.939791747211975e-05
err =  0.0013367526661749417
Newton iteration  10
Energy:  -7.037600646014415e-05
err =  0.001321937926707886
Newton iteration  11
Energy:  -7.129562630717556e-05
err =  0.0009065831016037766
Newton iteration  12
Energy:  -7.179495559648504e-05
err =  0.0011375996660445756
Newton iteration  13
Energy:  -7.239134625847884e-05
err =  0.0005871207552961564
Newton iteration  14
Energy:  -7.257405902486984e-05
err =  0.0003981407683686109
Newton iteration  15
Energy:  -7.257594084617175e-05
err =  0.0005631200806648349
Newton iteration  16
Energy:  -7.245165804309842e-05
err =  0.0008176865924930047
Newton iteration  17
Energy:  -7.27786612493251e-05
err =  0.0003463430715229245
Newton iteration  18
Energy:  -7.283197012199388e-05
err =  0.0002652353335713162
Newton iteration  19
Energy:  -7.287135604967931e-05
err =  0.00017090003131437992
Newton iteration  20
Energy:  -7.288486823434267e-05
err =  0.00017696973148516405
Newton iteration  21
Energy:  -7.289404779586066e-05
err =  0.00017302084860590475
Newton iteration  22
Energy:  -7.291048548422409e-05
err =  9.48691410459553e-05
Newton iteration  23
Energy:  -7.291531312499661e-05
err =  6.700572963859497e-05
Newton iteration  24
Energy:  -7.291803607927165e-05
err =  6.0549651764592904e-05
Newton iteration  25
Energy:  -7.2920088279602e-05
err =  2.9397870586293024e-05
Newton iteration  26
Energy:  -7.29205821859235e-05
err =  2.2224946982698688e-05
Newton iteration  27
Energy:  -7.292086834449779e-05
err =  1.189034072931015e-05
Newton iteration  28
Energy:  -7.29209427090438e-05
err =  5.2333012082606884e-06
Newton iteration  29
Energy:  -7.292095664456944e-05
err =  5.036213905569538e-07
Newton iteration  30
Energy:  -7.29209567859926e-05
err =  1.919791289273044e-07
Newton iteration  31
Energy:  -7.29209568044208e-05
err =  9.406202374902876e-16

[16]:

Draw (gfu, mesh, deformation=True);

[ ]: