AFEM 1.0.7

A finite element Python implementation

FEM

N dimensional FEM implementation for M variables per node problems.

Tutorial

Using pre implemented equations

Avaliable equations:

• 1D 1 Variable ordinary diferential equation
• 1D 2 Variable Euler Bernoulli Beams [TODO]
• 1D 2 Variable Timoshenko Beams [TODO]
• 2D 1 Variable Torsion
• 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
import FEM #import AFEM
from FEM import Mesh #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 = Mesh.Delaunay._strdelaunay(constrained=True, delaunay=True,
                                    a='0.00003', o=2)
#**Create** geometry using triangulation parameters. Geometry can be imported from .msh files.
geometry = Mesh.Delaunay1V(vertices, params)

#Save geometry to .msh file
geometry.saveMesh('I_test')

#Create torsional 2D analysis.
O = FEM.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
import FEM #import AFEM
from FEM import Mesh #Import Meshing tools

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

#Load geometry with file.
geometry = Mesh.Geometry.loadmsh('I_test.msh')

#Create torsional 2D analysis.
O = FEM.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. 1D and 2D heat transfer
4. Geometry class modification for hierarchy with 1D, 2D and 3D geometry child classes
5. Transient analysis (Core modification)
6. Elasticity in 3D (3D meshing and post process)
7. Non Lineal analysis for 1D equation (All cases)
8. Non Lineal for 2D equation (All cases)
9. UNIT TESTING
10. NUMERICAL VALIDATION
11. 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]

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