# Variable Thickness Sheet
In variable thickness sheets, you optimize 2D geometries in terms of the 
tickness of each element $\boldsymbol{t}$. Since for isotropic linear 
elasticity, the element stiffness matrix $\boldsymbol{K}_e$ scales like 
$\boldsymbol{K}_e \propto E_e t_e$, this is the same as taking the conventional SIMP 
interpolation for rescaling the materials Young's modulus $E_0$ to an 
effictive Young's modulus $E_{eff}$ via
```{math}
E_{eff,e} = E_{min} + (E_0 - E_{min}) \rho_e^k
```
with $k=1$ instead of usual $k\approx3$. We perform this optimization for the 
basic cantilever case of the famous 88 line code case, via these few lines 
```
from topoptlab.topology_optimization import main
from topoptlab.example_bc.lin_elast import cantilever_2d

# The real main driver
if __name__ == "__main__":
    # geometry input parameters
    nelx = 160
    nely = int(5/8 * nelx)
    # volume constraint
    volfrac = 0.4
    # filter radius
    rmin = 3/80 *nelx
    # only set to one for variable thickness sheets. Otherwise k=1 is nonsense!
    penal = 1.0
    # sensitivity filter
    ft = 0 
    # display on screen
    display = True
    # export vtk file
    export = True
    # write a log file
    write_log = True
    # run the actual optimization
    main(nelx=nelx, nely=nely, 
         volfrac=volfrac, 
         penal=penal, rmin=rmin, 
         ft=ft, # integer indicating the filter to be used
         filter_mode="matrix", # indicating how filtering is performed
         optimizer="oc", # use optimality criteria method
         bcs=cantilever_2d, # boundary conditions for cantilever in 2d
         output_kw = {"file": "cantilever_2d",
                      "display": display,
                      "export": export,
                      "write_log": write_log,
                      "profile": False,
                      "verbosity": 20, 
                      "output_movie": False})
```
Take the vtk-file "cantilever_2d.vtk" and open it in Paraview. In "Properties",
click on "Coloring" which should have "Solid Color" written right now. Select
xPhys instead. To get a different coloring, click "Edit color map", go directly 
under the transfer functions to "Select a colormap from default maps" 
and select "X Ray". Now you should see

```{image} /_static/cantilever-variablethickness-1.png
:align: center
:alt: Mesh numbering
:width: 600px
```

To convert this view to a variable thickness design, go to 
filters, choose *Extract Surface* and after this *Linear Cell Extrusion*.  
Go to the bottom of the *Properties* of the *Linear Cell Extrusion*, unselect 
"Use Color Palette For Background", click on *Background* and choose 
white. Now you should see the final design

```{image} /_static/cantilever-variablethickness-2.png
:align: center
:alt: Mesh numbering
:width: 600px
```