AFEM’s documentation#

Created by Arturo Rodriguez: da.rodriguezh@uniandes.edu.co

Contents:

Indices and tables#

Index

Module Index

Search Page

A Python FEM implementation.

N dimensional FEM implementation for M variables per node problems.

Viewer#

FEMViewer

Installation#

Use the package manager pip to install AFEM.

pip install AFEM

From source:

git clone https://github.com/ZibraMax/FEM
cd FEM
python -m venv .venv
python -m pip install build
python -m build
python -m pip install -e .[docs] # Basic instalation with docs

Contributing#

Pull requests are welcome. For major changes, please open an issue first to discuss what you would like to change.

Please make sure to update tests as appropriate.

Full Docs #

Tutorial#

Using pre implemented equations#

Avaliable equations:

1D 1 Variable ordinary diferential equation
1D 1 Variable 1D Heat with convective border
1D 2 Variable Euler Bernoulli Beams
1D 3 Variable Non-linear Euler Bernoulli Beams
2D 1 Variable Torsion
2D 1 Variable Poisson equation
2D 1 Variable second order PDE
2D 1 Variable 2D Heat with convective borders
2D 2 Variable Plane Strees
2D 2 Variable Plane Strees Orthotropic
2D 2 Variable Plane Strain
3D 3 variables per node isotropic elasticity

Numerical Validation:

[x] 1D 1 Variable ordinary diferential equation
[ ] 1D 1 Variable 1D Heat with convective border
[ ] 1D 2 Variable Euler Bernoulli Beams
[ ] 1D 3 Variable Non-linear Euler Bernoulli Beams
[x] 2D 1 Variable Torsion
[ ] 2D 1 Variable 2D Heat with convective borders
[x] 2D 2 Variable Plane Strees
[x] 2D 2 Variable Plane Strain

Steps:#

Create geometry
Create Border Conditions (Point and regions supported)
Solve!
For example: Example 2, Example 5, Example 11-14

Example without geometry file (Test 2):#

import matplotlib.pyplot as plt #Import libraries
from FEM.Torsion2D import Torsion2D #import AFEM Torsion class
from FEM.Geometry import Delaunay #Import Meshing tools

#Define some variables with geometric properties
a = 0.3
b = 0.3
tw = 0.05
tf = 0.05

#Define material constants
E = 200000
v = 0.27
G = E / (2 * (1 + v))
phi = 1 #Rotation angle

#Define domain coordinates
vertices = [
        [0, 0],
        [a, 0],
        [a, tf],
        [a / 2 + tw / 2, tf],
        [a / 2 + tw / 2, tf + b],
        [a, tf + b],
        [a, 2 * tf + b],
        [0, 2 * tf + b],
        [0, tf + b],
        [a / 2 - tw / 2, tf + b],
        [a / 2 - tw / 2, tf],
        [0, tf],
]

#Define triangulation parameters with `_strdelaunay` method.
params = Delaunay._strdelaunay(constrained=True, delaunay=True,
                                                                        a='0.00003', o=2)
#**Create** geometry using triangulation parameters. Geometry can be imported from .msh files.
geometry = Delaunay(vertices, params)

#Save geometry to .json file
geometry.exportJSON('I_test.json')

#Create torsional 2D analysis.
O = Torsion2D(geometry, G, phi)
#Solve the equation in domain.
#Post process and show results
O.solve()
plt.show()

Example with geometry file (Example 2):#

import matplotlib.pyplot as plt #Import libraries
from FEM.Torsion2D import Torsion2D #import AFEM
from FEM.Geometry import Geometry #Import Geometry tools

#Define material constants.
E = 200000
v = 0.27
G = E / (2 * (1 + v))
phi = 1 #Rotation angle

#Load geometry with file.
geometry = Geometry.importJSON('I_test.json')

#Create torsional 2D analysis.
O = Torsion2D(geometry, G, phi)
#Solve the equation in domain.
#Post process and show results
O.solve()
plt.show()

Creating equation classes#

Note: Don’t forget the docstring!

Steps#

Create a Python flie and import the libraries:
```
from .Core import *
from tqdm import tqdm
import numpy as np
import matplotlib.pyplot as plt
```
- Core: Solver
- Numpy: Numpy data
- Matplotlib: Matplotlib graphs
- Tqdm: Progressbars

Create a Python class with Core inheritance

   class PlaneStress(Core):
       def __init__(self,geometry,*args,**kargs):
       #Do stuff
       Core.__init__(self,geometry)

It is important to manage the number of variables per node in the input geometry.

Define the matrix calculation methods and post porcessing methods.
```
def elementMatrices(self):
def postProcess(self):
```

The elementMatrices method uses gauss integration points, so you must use the following structure:

for e in tqdm(self.elements,unit='Element'):
    _x,_p = e.T(e.Z.T) #Gauss points in global coordinates and Shape functions evaluated in gauss points
    jac,dpz = e.J(e.Z.T) #Jacobian evaluated in gauss points and shape functions derivatives in natural coordinates
    detjac = np.linalg.det(jac)
    _j = np.linalg.inv(jac) #Jacobian inverse
    dpx = _j @ dpz #Shape function derivatives in global coordinates
    for k in range(len(e.Z)): #Iterate over gauss points on domain
        #Calculate matrices with any finite element model
    #Assign matrices to element

A good example is the PlaneStress class in the Elasticity2D.py file.

Roadmap#

2D elastic plate theory
Transient analysis (Core modification)
Non-Lineal for 2D equation (All cases)
Testing and numerical validation (WIP)

Examples #

References#

1. Reddy. Introduction to the Finite Element Method, Third Edition (McGraw-Hill Education: New York, Chicago, San Francisco, Athens, London, Madrid, Mexico City, Milan, New Delhi, Singapore, Sydney, Toronto, 2006). https://www.accessengineeringlibrary.com/content/book/9780072466850

Jonathan Richard Shewchuk, (1996) Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator

Ramirez, F. (2020). ICYA 4414 Modelación con Elementos Finitos [Class handout]. Universidad de Los Andes.

License#

MIT