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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 = 0.00017982731508198267
CG iteration 2, residual = 7.527505823604927e-05
CG iteration 3, residual = 8.667360647586832e-05
CG iteration 4, residual = 6.611763953432749e-05
CG iteration 5, residual = 6.0095304950833594e-05
CG iteration 6, residual = 4.240151181641312e-05
CG iteration 7, residual = 3.174659664109265e-05
CG iteration 8, residual = 2.7007176859764085e-05
CG iteration 9, residual = 2.0093499070754506e-05
CG iteration 10, residual = 1.62027846525673e-05
CG iteration 11, residual = 1.116424103583484e-05
CG iteration 12, residual = 8.707361209404812e-06
CG iteration 13, residual = 6.263300218968306e-06
CG iteration 14, residual = 4.8276622278931326e-06
CG iteration 15, residual = 3.504830182391182e-06
CG iteration 16, residual = 2.674487250730764e-06
CG iteration 17, residual = 1.995962625386408e-06
CG iteration 18, residual = 1.4173939614999919e-06
CG iteration 19, residual = 1.07359294206457e-06
CG iteration 20, residual = 7.342807879185708e-07
CG iteration 21, residual = 6.364131834893672e-07
CG iteration 22, residual = 4.130307829075218e-07
CG iteration 23, residual = 3.667583721055648e-07
CG iteration 24, residual = 2.5942494636493067e-07
CG iteration 25, residual = 1.8056292944582968e-07
CG iteration 26, residual = 1.3642541682195363e-07
CG iteration 27, residual = 9.965383340157184e-08
CG iteration 28, residual = 7.638450216779991e-08
CG iteration 29, residual = 5.2824180300395505e-08
CG iteration 30, residual = 4.016696599502561e-08
CG iteration 31, residual = 2.8530509651531822e-08
CG iteration 32, residual = 2.171386098777403e-08
CG iteration 33, residual = 1.695593122041116e-08
CG iteration 34, residual = 1.1309617391623708e-08
CG iteration 35, residual = 8.114589692955136e-09
CG iteration 36, residual = 5.851886240187138e-09
CG iteration 37, residual = 4.026669356688742e-09
CG iteration 38, residual = 3.0617221443683202e-09
CG iteration 39, residual = 2.5243980127966064e-09
CG iteration 40, residual = 1.6419933510964724e-09
CG iteration 41, residual = 1.1584804532336763e-09
CG iteration 42, residual = 9.165977609891173e-10
CG iteration 43, residual = 6.363596328349435e-10
CG iteration 44, residual = 4.1915904050300315e-10
CG iteration 45, residual = 4.0370281339198724e-10
CG iteration 46, residual = 2.7124455381716286e-10
CG iteration 47, residual = 1.9147391700882495e-10
CG iteration 48, residual = 1.459837443152833e-10
CG iteration 49, residual = 1.209772807630053e-10
CG iteration 50, residual = 9.37062575374736e-11
CG iteration 51, residual = 6.578121723843675e-11
CG iteration 52, residual = 4.1286567270040894e-11
CG iteration 53, residual = 3.0961838788596964e-11
CG iteration 54, residual = 2.025611286831344e-11
CG iteration 55, residual = 1.5070685002065613e-11
CG iteration 56, residual = 1.1829537820308014e-11
CG iteration 57, residual = 7.406491495862401e-12
CG iteration 58, residual = 5.294041051469543e-12
CG iteration 59, residual = 3.45336319937333e-12
CG iteration 60, residual = 2.3843318387792995e-12
CG iteration 61, residual = 1.6369579396203232e-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);