3D Solid Mechanics

1. 3D Solid Mechanics#

  • Open Cascade Techonology geometry kernel

  • Netgen mesh generator

  • NGSolve Finite Element libraray

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

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

ea = { "euler_angles" : [-70,5,30] }
Draw(geo, **ea);

find edges between box and cylinder, and build chamfers:

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"

Draw(geo, **ea);

mesh = Mesh(OCCGeometry(geo).GenerateMesh(maxh=0.1)).Curve(3)
Draw (mesh, **ea);

1.1. Linear elasticity#

Displacement: \(u : \Omega \rightarrow {\mathbb R}^3\)

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:

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

1.2. 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 = preconditioners.BDDC(a)
    a.Assemble()

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

inv = solvers.CGSolver(a.mat, pre, plotrates=True, tol=1e-8)
gfu.vec.data = inv * f.vec
../_images/436df5f5cf8bacf5dd91f09eaea086a4b787eff2bf47ce4882cac2e320d04e6f.png 
<Figure size 640x480 with 0 Axes>

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

Draw (gfu, mesh, deformation=True, scale=3e4, draw_vol=False, **ea);

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