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1D and 2D FULL FEM implementation
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@ZibraMax ZibraMax released this 21 Mar 03:42
· 399 commits to master since this release
v1.0.2
328f93c

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()

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

Assets 4
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