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3D Solid Mechanics

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

box = Box((0,0,0), (3,0.6,1))
box.faces.name="outer"
cyl = sum( [Cylinder((0.5+i,0,0.5), Y, 0.25,0.8) for i in range(3)] )
cyl.faces.name="cyl"
geo = box-cyl

DrawGeo(geo);

find edges between box and cylinder, and build chamfers (requires OCC 7.4 or newer):

cylboxedges = geo.faces["outer"].edges * geo.faces["cyl"].edges
cylboxedges.name = "cylbox"
geo = geo.MakeChamfer(cylboxedges, 0.03)

name faces for boundary conditions:

geo.faces.Min(X).name = "fix"
geo.faces.Max(X).name = "force"

DrawGeo(geo);

from ngsolve import *
from ngsolve.webgui import Draw
mesh = Mesh(OCCGeometry(geo).GenerateMesh(maxh=0.1)).Curve(3)
Draw (mesh);

Linear elasticity

Displacement: \(u : \Omega \rightarrow R^3\)

Linear strain:

\[\varepsilon(u) := \tfrac{1}{2} ( \nabla u + (\nabla u)^T )\]

Stress by Hooke’s law:

\[\sigma = 2 \mu \varepsilon + \lambda \operatorname{tr} \varepsilon I\]

Equilibrium of forces:

\[\operatorname{div} \sigma = f\]

Displacement boundary conditions:

\[u = u_D \qquad \text{on} \, \Gamma_D\]

Traction boundary conditions:

\[\sigma n = g \qquad \text{on} \, \Gamma_N\]

Variational formulation:

Find: \(u \in H^1(\Omega)^3\) such that \(u = u_D\) on \(\Gamma_D\)

\[\int_\Omega \sigma(\varepsilon(u)) : \varepsilon(v) \, dx = \int_\Omega f v dx + \int_{\Gamma_N} g v ds\]

holds for all \(v = 0\) on \(\Gamma_D\).

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

def Stress(strain):
    return 2*mu*strain + lam*Trace(strain)*Id(3)

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

with TaskManager():
    a = BilinearForm(InnerProduct(Stress(Sym(Grad(u))), Sym(Grad(v))).Compile()*dx)
    pre = Preconditioner(a, "bddc")
    a.Assemble()

force = CF( (1e-3,0,0) )
f = LinearForm(force*v*ds("force")).Assemble()

from ngsolve.krylovspace import CGSolver
inv = CGSolver(a.mat, pre, printrates=True, tol=1e-8)
gfu.vec.data = inv * f.vec

CG iteration 1, residual = 0.00017982731508198229
CG iteration 2, residual = 7.527505823605007e-05

CG iteration 3, residual = 8.667360647586976e-05
CG iteration 4, residual = 6.611763953432795e-05

CG iteration 5, residual = 6.009530495083395e-05
CG iteration 6, residual = 4.240151181641361e-05

CG iteration 7, residual = 3.174659664109286e-05
CG iteration 8, residual = 2.7007176859763906e-05

CG iteration 9, residual = 2.009349907075462e-05
CG iteration 10, residual = 1.6202784652567423e-05

CG iteration 11, residual = 1.1164241035835019e-05
CG iteration 12, residual = 8.707361209404916e-06

CG iteration 13, residual = 6.263300218968382e-06
CG iteration 14, residual = 4.827662227893158e-06

CG iteration 15, residual = 3.504830182391232e-06
CG iteration 16, residual = 2.6744872507307964e-06

CG iteration 17, residual = 1.9959626253864205e-06
CG iteration 18, residual = 1.4173939615000052e-06

CG iteration 19, residual = 1.0735929420645797e-06
CG iteration 20, residual = 7.342807879185295e-07

CG iteration 21, residual = 6.364131834892673e-07
CG iteration 22, residual = 4.130307829073085e-07

CG iteration 23, residual = 3.6675837210550135e-07
CG iteration 24, residual = 2.594249463649674e-07

CG iteration 25, residual = 1.8056292944584655e-07
CG iteration 26, residual = 1.364254168219582e-07

CG iteration 27, residual = 9.965383340157333e-08
CG iteration 28, residual = 7.638450216780137e-08

CG iteration 29, residual = 5.282418030039682e-08
CG iteration 30, residual = 4.0166965995026514e-08

CG iteration 31, residual = 2.8530509651532163e-08
CG iteration 32, residual = 2.1713860987774794e-08

CG iteration 33, residual = 1.6955931220412304e-08
CG iteration 34, residual = 1.1309617391628812e-08

CG iteration 35, residual = 8.11458969299672e-09
CG iteration 36, residual = 5.851886240699253e-09

CG iteration 37, residual = 4.026669362860067e-09
CG iteration 38, residual = 3.0617222125898965e-09

CG iteration 39, residual = 2.5243982340915296e-09
CG iteration 40, residual = 1.6419943106258823e-09

CG iteration 41, residual = 1.158492046813089e-09
CG iteration 42, residual = 9.16688794388909e-10

CG iteration 43, residual = 6.368216008850478e-10
CG iteration 44, residual = 4.264327189487429e-10

CG iteration 45, residual = 4.2993994837012813e-10
CG iteration 46, residual = 2.633992837500192e-10

CG iteration 47, residual = 1.9050949561930513e-10
CG iteration 48, residual = 1.4244695711205855e-10

CG iteration 49, residual = 1.0324792836542466e-10
CG iteration 50, residual = 8.717497360795928e-11

CG iteration 51, residual = 7.17384786495039e-11
CG iteration 52, residual = 4.1580100070415294e-11

CG iteration 53, residual = 3.097097130804017e-11
CG iteration 54, residual = 2.0256355184887157e-11

CG iteration 55, residual = 1.5070691975664183e-11
CG iteration 56, residual = 1.1829538382473839e-11

CG iteration 57, residual = 7.406491647801214e-12
CG iteration 58, residual = 5.294041938673523e-12

CG iteration 59, residual = 3.4533630559091152e-12
CG iteration 60, residual = 2.3841403522357644e-12

CG iteration 61, residual = 1.6294413461680155e-12

with TaskManager():
    fesstress = MatrixValued(H1(mesh,order=3), symmetric=True)
    gfstress = GridFunction(fesstress)
    gfstress.Interpolate (Stress(Sym(Grad(gfu))))

Draw (5e4*gfu, mesh);

Draw (Norm(gfstress), mesh, deformation=1e4*gfu, draw_vol=False, order=3);