Anisotropic Materials in 3D Printing

[Preetzl]

Hello kinnala,

Thank you for creating a very nice library! I was wondering if it is possible to implement a different elasticity module for problems like ex11 as I think that this is what you need to simulate 3d printed parts.

Kind regards

[kinnala]

If I understand correctly, you want to have a different elastic modulus for the different parts of the domain.

Yes, it is possible to define the elastic modulus element-by-element. Since I don't know exactly what you have in mind, perhaps you can elaborate it with a drawing?

[Preetzl]

Thank you for your fast responds. I still couldn`t figure it out to implement the anisotropic elasticity form. I want to try the following steps described in the post: https://en.wikipedia.org/wiki/Transverse_isotropy.

The stiffness matrix for a simple cube is given there. Since I'm clearly too inexperienced in programming and also don't quite understand how you make your composition of the stiffness matrix, I wanted to ask how you would do it. Is it even possible to keep it that simple?

Sorry for the wrong question at the beginning and the late answer.

[kinnala]

I misunderstood your question completely, I thought you want to simply have a different elasticity modulus for the different layers but still consider an isotropic material. However, now I think that what you want to do is to specify a general elasticity 4-tensor $C_{ijkl}$. This should be an easy modification of the variational form. In general, the bilinear form for linear elasticity is
$$\int_\Omega C_{ijkl} \varepsilon_{ij}(u) \varepsilon_{kl}(v) dx.$$
Now we simply need to implement this bilinear form in code.

[kinnala]

I don't like the Voigt notation, I see it as an unnecessary mathematical trick to turn the contraction between 4-tensor $C$ and 2-tensor $\varepsilon$ into a matrix-vector product which people might sometimes be more familiar with. However, if you want to use it, you can then do the mapping from and to Voigt notation.

[kinnala]

Anyhow, if you want to define a general 4-tensor in numpy, you can do it as follows:

import numpy as np
# initialize 4-tensor
C = np.zeros((3, 3, 3, 3))
# set correct values to all of its components
C[0, 0, 0, 0] = ...
C[0, 0, 0, 1] = ...

That should be done according to your anisotropic material model.

[kinnala]

The next step is to implement the general bilinear form, I think the following should be enough for all cases:

from skfem import *
from skfem.helpers import *

# assuming C is defined as above ...

@BilinearForm
def anisotropic(u, v, _):
    epsu = 0.5 * (grad(u) + transpose(grad(u)))
    epsv = 0.5 * (grad(v) + transpose(grad(v)))
    return np.einsum('ijkl,ij...,kl...', C, epsu, epsv)