AFEM 1.0.24

A finite element Python implementation

A Python FEM implementation.

N dimensional FEM implementation for M variables per node problems.

Installation

Use the package manager pip to install AFEM.

pip install AFEM

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 [TODO]
• 1D 2 Variable Timoshenko Beams [TODO]
• 2D 1 Variable Torsion
• 2D 1 Variable 2D Heat with convective borders
• 2D 2 Variable Plane Strees
• 2D 2 Variable Plane Strain

Steps:

• Create geometry (From coordinates or GiD)
• Create Border Conditions (Point and segment supported)
• Solve!
• For example: Test 2, Test 5, Test 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.Mesh.Delaunay 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 .msh file
geometry.saveMesh('I_test')

#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 (Test 2):

import matplotlib.pyplot as plt #Import libraries
from FEM.Torsion2D import Torsion2D #import AFEM
from FEM.Mesh.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.loadmsh('I_test.msh')

#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

1. 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
   • Core: Numpy data
   • Core: Matplotlib graphs
   • Tqdm: Progressbars

2. 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.

3. Define the matrix calculation methods and post porcessing methods.

   def elementMatrices(self):
   def postProcess(self):
   

4. 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

Roadmap

1. Beam bending by Euler Bernoulli and Timoshenko equations
2. 2D elastic plate theory
3. Geometry class modification for hierarchy with 1D, 2D and 3D geometry child classes
4. Transient analysis (Core modification)
5. Elasticity in 3D (3D meshing and post process)
6. Non-Lineal analysis for 1D equation (All cases)
7. Non-Lineal for 2D equation (All cases)
8. UNIT TESTING
9. NUMERICAL VALIDATION
10. Non-Local 2D?

Test index:

• Test 1: Preliminar geometry test

• Test 2: 2D Torsion 1 variable per node. H section-Triangular Quadratic.

  

• Test 3: 2D Torsion 1 variable per node. Square section-Triangular Quadratic.

  

• Test 4: 2D Torsion 1 variable per node. Mesh from internet-Square Lineal.

  

• Test 5: 2D Torsion 1 variable per node. Creating and saving mesh-Triangular Quadratic.

  

• Test 6: 1D random differential equation 1 variable per node. Linear Quadratic.

  

• Test 7: GiD Mesh import test — Serendipity elements

  

• Test 8: Plane Stress 2 variable per node. Plate in tension — Serendipity.

  

• Test 9: Plane Stress 2 variable per node. Simple Supported Beam — Serendipity.

  

• Test 10: Plane Stress 2 variable per node. Cantilever Beam — Triangular Quadratic.

  

• Test 11: Plane Stress 2 variable per node. Fixed-Fixed Beam — Serendipity.

  

• Test 12: Plane Strain 2 variable per node. Embankment from GiD — Serendipity.

  

• Test 13: Plane Strain 2 variable per node. Embankment — Triangular Quadratic.

  

• Test 14: Plane Stress 2 variable per node. Cantilever Beam — Serendipity.

  

• Test 15: Profile creation tool. Same as Test 14

  

• Test 16: Non-Local Plane Stress. [WIP]

• Test 17: 1D Heat transfer.

  

• Test 18: 2D border elements creation.

  

• Test 19: Apply loads on segments. loadOnSegment method on Test 11

  

• Test 20: Reddy's Example 11.7.1 Ed 3

• Test 21: Test 20 with serendipity elements.

• Test 22: Test 20 with refined mesh.

  

• Test 23: Reddy's Problem 11.1 Ed 3 Plain Strain

  

• Test 24: Test 23 with refined mesh

  

• Test 25: Holes concept. With Test 24

  

• Test 26: Fillets concept.

  

• Test 27: Combination of Holes Fillets, Plane Stress

  

• Test 28: Fillets and Holes mesh files of Test 27

  

• Test 29: Fillets and Holes in Test 13

  

• Test 30: Border conditions and loads in holes

  

• Test 31: 2D Heat with convective borders

  

• Test 32: Border conditions and loads in holes

  

• Test 33: Test 30 with Heat

  

• Test 34: Custom plots, Beam-Girder steel plate connection

  

References

J. N. 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

License

MIT