inspired by the im2mesh matlab code.

Images to mesh

FEM simulation

FEnics on win

Fenics is primary developed for linux and macos, for win, insall use docer

1. install docker
2. install fenics iamges 'docker pull quay.io/fenicsproject/stable:current'
3. Mounting Local Directories 'path' and Jupyter notebook:

docker run -ti --rm -p 8888:8888 -v llocal_directories_to_mount:/home/fenics/shared quay.io/fenicsproject/stable:current 'jupyter-notebook --ip=0.0.0.0 --port=8888 --no-browser --allow-root' 4. lauch jupyter notebook: http://127.0.0.1:8888/?token=your_token

Arc length solution for D-F

arc legnth solution

FEniCs math

linear problem

$\text{Find } \boldsymbol{u}\in V \text{ s.t. } \int_{\Omega} \boldsymbol{\sigma}(\boldsymbol{u}):\boldsymbol{\varepsilon}(\boldsymbol{v}) d\Omega = \int_{\Omega} \boldsymbol{f}\cdot\boldsymbol{v} d\Omega \quad \forall\boldsymbol{v} \in V $

V = VectorFunctionSpace(mesh, "Lagrange", 1)
u = TrialFunction(V)
v = TestFunction(V)
lsh = dot(inner(sigma(u), eps(v))*dx)
rsh = dot(f,u)*dx

weak coupled thmermomechancial problem

\begin{equation} \boldsymbol{\sigma} = \mathbb{C}:(\boldsymbol{\varepsilon}-\alpha(T-T_0)\boldsymbol{1}) = \lambda\text{tr}(\boldsymbol{\varepsilon})\boldsymbol{1}+2\mu\boldsymbol{\varepsilon} -\alpha(3\lambda+2\mu)(T-T_0)\boldsymbol{1} \end{equation}

V = VectorFunctionSpace(mesh, 'CG', 2)
u = TrialFunction(V)
v = TestFunction(V)
Wint = inner(sigma(u, Delta_T), eps(v))*dx
aM = lhs(Wint)
LM = rhs(Wint) + inner(f, v)*dx
bcu = DirichletBC(V, Constant((0., 0.)), lateral_sides)
u = Function(V, name="Displacement")
solve(aM == LM, u, bcu)