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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: $u:\mathrm{\Omega }\to R^3$

Linear strain:

$$\epsilon (u):=\frac{1}{2}(\mathrm{\nabla }u+(\mathrm{\nabla }u{)}^T)$$

Stress by Hooke’s law:

$$\sigma =2\mu \epsilon +\lambda \mathrm{tr}\epsilon I$$

Equilibrium of forces:

$$\mathrm{div}\sigma =f$$

Displacement boundary conditions:

$$u=u_D\text{on}{\mathrm{\Gamma }}_D$$

Traction boundary conditions:

$$\sigma n=g\text{on}{\mathrm{\Gamma }}_N$$

Variational formulation:

Find: $u\in H^1(\mathrm{\Omega }{)}^3$ such that $u=u_D$ on ${\mathrm{\Gamma }}_D$

$${\int }_{\mathrm{\Omega }}\sigma (\epsilon (u)):\epsilon (v)dx={\int }_{\mathrm{\Omega }}fvdx+{\int }_{{\mathrm{\Gamma }}_N}gvds$$

holds for all $v=0$ on ${\mathrm{\Gamma }}_D$.

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 = 0.00017982731508198196
CG iteration 2, residual = 7.527505823604999e-05
CG iteration 3, residual = 8.667360647586854e-05
CG iteration 4, residual = 6.611763953432786e-05
CG iteration 5, residual = 6.009530495083377e-05
CG iteration 6, residual = 4.240151181641339e-05
CG iteration 7, residual = 3.174659664109285e-05
CG iteration 8, residual = 2.7007176859764265e-05
CG iteration 9, residual = 2.0093499070754638e-05
CG iteration 10, residual = 1.620278465256741e-05
CG iteration 11, residual = 1.1164241035834938e-05
CG iteration 12, residual = 8.707361209404863e-06
CG iteration 13, residual = 6.263300218968363e-06
CG iteration 14, residual = 4.827662227893153e-06
CG iteration 15, residual = 3.5048301823912414e-06
CG iteration 16, residual = 2.6744872507307778e-06
CG iteration 17, residual = 1.9959626253864358e-06
CG iteration 18, residual = 1.4173939615000084e-06
CG iteration 19, residual = 1.0735929420645687e-06
CG iteration 20, residual = 7.342807879185329e-07
CG iteration 21, residual = 6.364131834892706e-07
CG iteration 22, residual = 4.1303078290730905e-07
CG iteration 23, residual = 3.667583721055049e-07
CG iteration 24, residual = 2.5942494636496794e-07
CG iteration 25, residual = 1.805629294458435e-07
CG iteration 26, residual = 1.3642541682195863e-07
CG iteration 27, residual = 9.965383340157361e-08
CG iteration 28, residual = 7.638450216780138e-08
CG iteration 29, residual = 5.282418030039626e-08
CG iteration 30, residual = 4.016696599502615e-08
CG iteration 31, residual = 2.853050965153251e-08
CG iteration 32, residual = 2.171386098777419e-08
CG iteration 33, residual = 1.6955931220410742e-08
CG iteration 34, residual = 1.1309617391619725e-08
CG iteration 35, residual = 8.114589692920307e-09
CG iteration 36, residual = 5.8518862397539e-09
CG iteration 37, residual = 4.02666935143064e-09
CG iteration 38, residual = 3.061722085798082e-09
CG iteration 39, residual = 2.524397820390125e-09
CG iteration 40, residual = 1.6419925053863179e-09
CG iteration 41, residual = 1.158470124444963e-09
CG iteration 42, residual = 9.165155677560578e-10
CG iteration 43, residual = 6.359342431270262e-10
CG iteration 44, residual = 4.120788555112914e-10
CG iteration 45, residual = 3.475360816124547e-10
CG iteration 46, residual = 2.913862570760715e-10
CG iteration 47, residual = 1.92728323632225e-10
CG iteration 48, residual = 1.4408886051090374e-10
CG iteration 49, residual = 1.122426507311188e-10
CG iteration 50, residual = 9.517641882334069e-11
CG iteration 51, residual = 6.691289015672579e-11
CG iteration 52, residual = 4.132646365973799e-11
CG iteration 53, residual = 3.0963083090874015e-11
CG iteration 54, residual = 2.025614648097733e-11
CG iteration 55, residual = 1.5070686043116477e-11
CG iteration 56, residual = 1.1829538190137948e-11
CG iteration 57, residual = 7.406493281312491e-12
CG iteration 58, residual = 5.294067613357582e-12
CG iteration 59, residual = 3.453776593803619e-12
CG iteration 60, residual = 2.390872304120967e-12
CG iteration 61, residual = 1.7659815586180077e-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);