PythonFem

<img src="./gifs/Multiple_2D_Neohookean_implicit_False_bend_8.gif" div align=middle width = "49%" /><img src="./gifs/Multiple_2D_Neohookean_implicit_False_fall_8.gif" div align=middle width = "49%" />

Linear Finite Element Method (FEM) for simulating hyperelastic cubes, all codes are based on pure Python.

Overview

| Constitutive Models | Time Integration | Multiple cubes | | :------------- | :----------: | :----------: | | Linear Elasticity | Symplectic Euler | Yes | | St. Venant-Kirchhoff | Symplectic/Implict Euler | Yes | | Corotated linear elasticity | Symplectic Euler | Yes | | Neohookean elasticity | Symplectic/Implict Euler | Yes |

Requirements

imageio==2.28.1
matplotlib==3.5.3
numpy==1.21.5
seaborn==0.12.2

Usage

Single

python main_single_2d.py

Args

# Parameters of Afine transformation
'--theta', type=float, default=-25.0, help='Counterclockwise rotation angle'
'--tx', type=float, default=3.0, help='Movement along the x-axis'
'--ty', type=float, default=3.0, help='Movement along the y-axis'
'--sx', type=float, default=2, help='Stretching along the x-axis'
'--sy', type=float, default=2, help='Stretching along the y-axis'
    
# Parameters of FEM
'--model', type=int, default=3, help='Constitutive model, 0:Linear, 1:STVK, 2:Co-rotated, 3: Neohookean'
'--Y', type=int, default=100, help='Youngs modulus'
'--V', type=float, default=0.1, help='Poisson ratio, [-1, 0.5]'
'--dt', type=float, default=0.01, help='Time horizon'
'--max_iter', type=int, default=200, help='Iteration'
'--mass', type=float, default=1.0, help='Mass of the point'
'--gravity', type=bool, default=True, help='Add gravity'
'--fix', type=bool, default=True, help='Fix a point on the wall' 
'--implict', action='store_true', help='Implict Euler Method, for STVK and Neohookean'

Visual Results

<p align="center">Symplectic Euler Methods</p>

<img src="./gifs/single_2D_Linear_implicit_False.gif" div align=middle width = "49%" /><img src="./gifs/single_2D_STVK_implicit_False.gif" div align=middle width = "49%" />

<img src="./gifs/single_2D_Co-rotated_implicit_False.gif" div align=middle width = "49%" /><img src="./gifs/single_2D_Neohookean_implicit_False.gif" div align=middle width = "49%" />

<p align="center">Implict Euler Methods</p>

<img src="./gifs/single_2D_STVK_implicit_True.gif" div align=middle width = "49%" /><img src="./gifs/single_2D_Neohookean_implicit_True.gif" div align=middle width = "49%" />

Multiple

python main_multiple_2d.py

Args

'--voxelx', type=int, default=8, help='Number of voxels (x-axis)'
'--voxely', type=int, default=3, help='Number of voxels (y-axis)'
'--em', type=list, default=[2,3,6,7,10,11,14,15], help='Empty voxels:[],[2,3,6,7,10,11,14,15],[2,3,6,7]'
    
# Parameters of FEM
'--mode', type=str, default="bend", help='Mode: bend/fall'
'--model', type=int, default=0, help='Constitutive model, 0:Linear, 1:STVK, 2:Co-rotated, 3: Neohookean'
'--Y', type=int, default=10000, help='Youngs modulus'
'--V', type=float, default=0.1, help='Poisson ratio [-1, 0.5]'
'--dt', type=float, default=0.001, help='Time horizon'
'--max_iter', type=int, default=200, help='Iteration'
'--mass', type=float, default=1.0, help='Mass of the point'
'--implict', action='store_true', help='Implict Euler Method, for STVK and Neohookean'

Visual Results

<p align="center"> Symplectic Euler Methods </p>

<p align="center"> Linear Elasticity </p>

<img src="./gifs/Multiple_2D_Linear_implicit_False_bend_8.gif" div align=middle width = "49%" /><img src="./gifs/Multiple_2D_Linear_implicit_False_fall_8.gif" div align=middle width = "49%" />

<p align="center"> St. Venant-Kirchhoff </p>

<img src="./gifs/Multiple_2D_STVK_implicit_False_bend_8.gif" div align=middle width = "49%" /><img src="./gifs/Multiple_2D_STVK_implicit_False_fall_8.gif" div align=middle width = "49%" />

<p align="center"> Corotated linear elasticity </p>

<img src="./gifs/Multiple_2D_Co-rotated_implicit_False_bend_8.gif" div align=middle width = "49%" /><img src="./gifs/Multiple_2D_Co-rotated_implicit_False_fall_8.gif" div align=middle width = "49%" />

<p align="center"> Neohookean elasticity </p>

<img src="./gifs/Multiple_2D_Neohookean_implicit_False_bend_8.gif" div align=middle width = "49%" /><img src="./gifs/Multiple_2D_Neohookean_implicit_False_fall_8.gif" div align=middle width = "49%" />

<p align="center"> Implict Euler Methods </p>

<p align="center"> St. Venant-Kirchhoff </p>

<img src="./gifs/Multiple_2D_STVK_implicit_True_bend_8.gif" div align=middle width = "49%" /><img src="./gifs/Multiple_2D_STVK_implicit_True_fall_8.gif" div align=middle width = "49%" />

<p align="center"> Neohookean elasticity </p>

<img src="./gifs/Multiple_2D_Neohookean_implicit_True_bend_8_inv.gif" div align=middle width = "49%" /><img src="./gifs/Multiple_2D_Neohookean_implicit_True_fall_8.gif" div align=middle width = "49%" />

Reference

FEM Simulation of 3D Deformable Solids: A practitioner's guide to theory, discretization and model reduction