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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 = 0.0001798904505629889
CG iteration 2, residual = 7.593089644801502e-05
CG iteration 3, residual = 9.553190911073905e-05
CG iteration 4, residual = 6.915155168636513e-05
CG iteration 5, residual = 6.408034527301711e-05
CG iteration 6, residual = 4.695723193578816e-05
CG iteration 7, residual = 4.074247131805326e-05
CG iteration 8, residual = 2.9643510454950516e-05
CG iteration 9, residual = 2.318959670509492e-05
CG iteration 10, residual = 1.72425409848226e-05
CG iteration 11, residual = 1.3840252847566131e-05
CG iteration 12, residual = 1.1371290859109169e-05
CG iteration 13, residual = 7.557236098508671e-06
CG iteration 14, residual = 5.772464004177653e-06
CG iteration 15, residual = 4.233341381713314e-06
CG iteration 16, residual = 3.46578446072407e-06
CG iteration 17, residual = 2.3625074117510964e-06
CG iteration 18, residual = 1.9603349624428e-06
CG iteration 19, residual = 1.4687989703180504e-06
CG iteration 20, residual = 1.1178053345310865e-06
CG iteration 21, residual = 7.786087109406234e-07
CG iteration 22, residual = 5.912354552243846e-07
CG iteration 23, residual = 4.871534255435932e-07
CG iteration 24, residual = 3.1939698996758633e-07
CG iteration 25, residual = 2.64489949154773e-07
CG iteration 26, residual = 1.7131785458603425e-07
CG iteration 27, residual = 1.5039339583039745e-07
CG iteration 28, residual = 1.0180139619024902e-07
CG iteration 29, residual = 7.508935450463243e-08
CG iteration 30, residual = 5.48559182751504e-08
CG iteration 31, residual = 4.2398287558211404e-08
CG iteration 32, residual = 2.787562233980415e-08
CG iteration 33, residual = 2.1924668675138876e-08
CG iteration 34, residual = 1.612943180976563e-08
CG iteration 35, residual = 1.1134506189519913e-08
CG iteration 36, residual = 8.848793325050694e-09
CG iteration 37, residual = 5.85370418819657e-09
CG iteration 38, residual = 4.719929037268229e-09
CG iteration 39, residual = 3.069207332782225e-09
CG iteration 40, residual = 2.281544468872042e-09
CG iteration 41, residual = 1.5354409421724715e-09
CG iteration 42, residual = 1.2608631037885828e-09
CG iteration 43, residual = 1.0526549824128506e-09
CG iteration 44, residual = 8.935353847399173e-10
CG iteration 45, residual = 5.071423911367238e-10
CG iteration 46, residual = 3.981818349060081e-10
CG iteration 47, residual = 3.0257296642966846e-10
CG iteration 48, residual = 2.0631083361852435e-10
CG iteration 49, residual = 1.7447750895645804e-10
CG iteration 50, residual = 1.5179119743771465e-10
CG iteration 51, residual = 9.463733920899645e-11
CG iteration 52, residual = 6.956163676029495e-11
CG iteration 53, residual = 5.1626792097454164e-11
CG iteration 54, residual = 5.0302888814501675e-11
CG iteration 55, residual = 2.7793127648769823e-11
CG iteration 56, residual = 1.985490056742012e-11
CG iteration 57, residual = 1.4157826454219715e-11
CG iteration 58, residual = 1.049955157296758e-11
CG iteration 59, residual = 7.383191335676521e-12
CG iteration 60, residual = 4.9347735880601855e-12
CG iteration 61, residual = 4.168146858608555e-12
CG iteration 62, residual = 2.749075473289132e-12
CG iteration 63, residual = 1.871879559034848e-12
CG iteration 64, residual = 1.928898118449263e-12
CG iteration 65, residual = 1.0739806882220427e-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);