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cubrium-0.1.5
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Cube in Equilibrium
| ویژگی | مقدار |
|---|---|
| سیستم عامل | - |
| نام فایل | cubrium-0.1.5 |
| نام | cubrium |
| نسخه کتابخانه | 0.1.5 |
| نگهدارنده | [] |
| ایمیل نگهدارنده | [] |
| نویسنده | Andreas Dutzler |
| ایمیل نویسنده | a.dutzler@gmail.com |
| آدرس صفحه اصلی | https://github.com/adtzlr/cubrium |
| آدرس اینترنتی | https://pypi.org/project/cubrium/ |
| مجوز | GPL-3.0-or-later |
# cubrium
A **cub**e in equilib**rium**
[](https://pypi.python.org/pypi/cubrium/)
[](https://github.com/psf/black)
<img src="https://raw.githubusercontent.com/adtzlr/cubrium/main/scripts/cube.png" width="75%">
Cubrium is a toolbox for the definition, solution and post-processing of homogenous loadcases in continuum mechanics of solids (statics). It uses [contique](https://github.com/adtzlr/contique/) for the numeric continuation of the nonlinear equilibrium equations. If you use `cubrium` or `contique` in your scientific publications, please cite my work. I'll provide a citable template with a DOI in the future.
[Official Documentation](https://adtzlr.github.io/cubrium/)
## Example 101 a.k.a `hello cubrium` 😎
This is an example which solves a cube with a Saint Venant-Kirchhoff (SVK) material for the case of uniaxial loading. In the first step, we `init` a model.
```python
import cubrium
MDL = cubrium.init()
```
### Constitution
In the second part, we have to define the constitutive law. We can either use one of `cubrium`'s models or use our own `umat` (user material). This time we're using our own `umat` function for a simple SVK material.
<img src="https://latex.codecogs.com/gif.latex?\boldsymbol{S}&space;=&space;2&space;\mu&space;\&space;\boldsymbol{E}&space;+&space;\gamma&space;\&space;\mathrm{tr}(\boldsymbol{E})&space;\&space;\boldsymbol{1}" title="\boldsymbol{S} = 2 \mu \ \boldsymbol{E} + \gamma \ \mathrm{tr}(\boldsymbol{E}) \ \boldsymbol{1}" />
```python
import numpy as np
def umat_svk(F, parameters):
"""(U)ser (MAT)erial Function.
Returns First Piola-Kirchhoff stress tensor for a given
deformation gradient tensor with a list of material parameters."""
# expand list of material parameters
mu, K = parameters[:2]
gamma = K - 2 / 3 * mu
C = F.T @ F
E = 1 / 2 * (C - np.eye(3))
S = 2 * mu * E + gamma * np.trace(E) * np.eye(3)
return F @ S
```
Now we have to link our `umat` to the `cubrium` model definition and specify material parameters.
```python
MDL.GLO.constitution.umat = umat_svk
MDL.GLO.constitution.parameters = [1.0, 5000.0]
```
### Loadcase (Kinematics and Kinetics)
A loadcase is defined with exactly **9** equations for the unsymmetric or **6** equations for the full-symmetric case. This contains either kinematic or kinetic types of equations. For the case of uniaxial loading we are building this loadcase for ourselfes. We apply an external normal force 1 and set all external shear forces and normal forces 2 and 3 to zero. A symmetric solution is enforced (no rigid body rotation is allowed). The load-proportionaly-factor is applied to the normal forces (`lpftype=0`). Finally we specify a `title` for the loadcase. This will later effect the output filenames.
```python
def uniaxial(MDL):
MDL.EXT.force.normal[0] = 1
MDL.EXT.force.normal[1] = 0
MDL.EXT.force.normal[2] = 0
MDL.EXT.force.shear[0, 1] = 0
MDL.EXT.force.shear[1, 2] = 0
MDL.EXT.force.shear[0, 2] = 0
MDL.EXT.gridvec.symmetry = [1, 1, 1]
MDL.GLO.lpftype = 0
MDL.GLO.title = "Uniaxial"
return MDL
```
Again, we have to link our loadcase to the `cubrium` model and update the model with the new loadcase settings.
```python
MDL = uniaxial(MDL)
MDL = cubrium.update(MDL)
```
### Solver
Starting from a valid initial solution everything is ready to solve the model. **Hint**: `x0` are the flattened components of the displacement gradient w.r.t. the undeformed coordinates (=primary unknows of the problem).
```python
Res = cubrium.solve(MDL)(
x0 = np.zeros(9),
lpf0 = 0.0,
)
```
The results contain the extended unknowns `y = (x, lpf)` but no information about the internal quantities of the model. Therefore we extract the extended unknows from the Result object (`Res`) and recover these internal quantities (e.g. reaction forces) for all steps.
```python
Y = np.array([res.x for res in Res])
history = cubrium.recover(Y, MDL)
```
### Plots and Post-processing
We plot the axial stretch vs. load-proportionality-factor in direction 1.
```python
import matplotlib.pyplot as plt
plt.plot([0], [0],"C0o", label="origin")
plt.plot(Y[:, 0], Y[:, -1], "C0.-")
plt.xlabel("$\lambda_1 - 1$")
plt.ylabel("load-proportionality-factor LPF")
```
<img src="https://raw.githubusercontent.com/adtzlr/cubrium/main/scripts/Uniaxial_stretch-normal-lpf.svg" width="75%">
Using `meshio` we are able to export our solution in the `xdmf` file format which may be further post-processed by ParaView.
```python
cubrium.writer.xdmf(
history,
filename = MDL.GLO.title,
)
```
An exemplary scene for ParaView 5.9.0 is available to [download](https://raw.githubusercontent.com/adtzlr/cubrium/main/scripts/paraviewstatecube.pvsm). Import it in ParaView (File - Load state) and choose "Choose File Names" as shown below. Voilà, a nice cube scene in 3D with a cube colored in "Cauchy Stress XX" and reaction forces scaled and colored in "Reaction Force Magnitude" is ready to animate. The whole script of this example may be [downloaded here](https://raw.githubusercontent.com/adtzlr/cubrium/main/scripts/script101_hellocubrium.py).
<img src="https://raw.githubusercontent.com/adtzlr/cubrium/main/scripts/paraviewimportscene.png" width="75%">
<img src="https://raw.githubusercontent.com/adtzlr/cubrium/main/scripts/paraviewcube.png" width="75%">
Finally you can watch the animated cube during the deformation.
<a href="ttps://raw.githubusercontent.com/adtzlr/cubrium/main/scripts/script101_hellocubrium_video.ogv"><img src="https://raw.githubusercontent.com/adtzlr/cubrium/main/scripts/script101_hellocubrium_video.gif" href="" width="75%"></a>
Have fun using `cubrium`! If you find any bugs please submit an issue.
نیازمندی
| مقدار | نام |
|---|---|
| - | numpy |
| - | contique |
| - | meshio |
زبان مورد نیاز
| مقدار | نام |
|---|---|
| >=3.6 | Python |
نحوه نصب
نصب پکیج whl cubrium-0.1.5:
pip install cubrium-0.1.5.whl
نصب پکیج tar.gz cubrium-0.1.5:
pip install cubrium-0.1.5.tar.gz