Finite Element Method

3D Demo: A simulation of an elasticity spot rendered using Houdini.

2D Demo: A simulation of an elasticity square mesh passing through a narrow passage.

A Finite Element Method (FEM) implemented in Taichi Lang that supports both 2D and 3D simulations. The FEM implementation includes support for both explicit and implicit integer methods. Have fun!

Feature

• Single object FEM simulation
• Hyperelastic material model: Neo-Hookean solid
• Explicit method:
  • Taichi autodiff system
  • Analytically differentiating
• Implicit method:
  • Jacobi iterative method
  • Conjugate gradient method
• Support *.obj and *.stl format 3D model input

Prerequisites

Taichi supports multiple difference backends, such as cuda, metal. For more detail, please refer to this. We utilize the pyvista, tetgen and trimesh packages to convert triangle meshes into tetrahedral meshes. The code test successfully passed on both Linux 22.04 and Windows 10, with CUDA as the backend. The necessary Python packages are listed below:

• taichi
• numpy
• pyvista
• tetgen
• trimesh

If you intend to output your simulation results in *.gif or *.mp4 format, it is necessary to have ffmpeg installed.

Usage

Quickstart

• Clone the repository

  git clone https://github.com/Jukgei/Finite-Element-Method.git

• Install the dependencies

  cd Finite-Element-Method
  pip install -r requirements.txt
  sudo apt install ffmpeg # if you intend to output your simulation results in `*.gif` of `*.mp4`

• Run simulation

  python main.py  # use default config

  If you are able to view the simulation, such as the 2D demo mentioned earlier, it indicates that your environment configuration is correct.
• Use custom config

  python main.py --config ./config/demo_3d.json 
  # Please change the "./config/demo_3d.json" to your config file path

Config explanation

The project utilizes the JSON format for its configuration files. The JSON config contains the following parameters:

• dim (integer): Specifies the dimension of the simulation. Valid values are 2 or 3.
• delta_time (float): Represents the step size of the simulation.
• sim_count (integer): Determines the number of simulation steps per render.
• auto_diff (boolean): Specifies whether to use the Taichi autodiff system. The parameter is only valid when "use_explicit_method": true
• use_explicit_method (boolean): Determines whether to use the explicit method for simulation, if set to false, the implicit method is used. If set to true, the explicit method is used.
• implicit_method (integer): Configures the solver for the implicit method. This parameter is only valid when "use_explicit_method": false. The following options are available:
  • 0: Jacobi iterative method
  • 1: Conjugate gradient method
• preconditioned (integer): Specifies whether to use the preconditioning technique before applying the conjugate gradient method to solve the system. This parameter is only valid when "use_explicit_method": false and implicit_method: 1. The following options are available:
  • 0: No use of the preconditioning technique.
  • 1: Use of the preconditioning technique with the equation $A^TAx = A^Tb$
• g_dir (list): Represents the gravity direction of the simulation. For example, [0, -1, 0] indicates gravity in the negative y-axis direction.
• is_output_gif (boolean): Determines whether to output the simulation process as a *.gif format result.
• is_output_obj (boolean): Determines whether to output the simulation process as a *.obj format result. This parameter is only valid in 2D simulation.
• output_fps (integer): Specifies the frames per second of the output.
• objects (list): Contains the simulation objects and their related parameters.
• blocks (list): Contains the simulation blocks and their related parameters. This parameter is only valid in 2D simulation. Currently only circular obstacles are supported.

Each object within the objects list contains the following parameters:

• id (integer): Represents the object's ID.
• rho (integer): Specifies the density of the object.
• center: Represents the center of the object.
• side_length (float) (2D only): Represents the side length of the simulation square.
• subdivisions (integer) (2D only): Represents the subdivisions of each side of the square.
• E (integer): Specifies the Young's modulus of the object.
• nu (float): Specifies the Poisson ratio of the object.
• damping (float): Specifies the simulation damping of the object.
• obj (string) (3D only): Specifies the path of the simulation object.

Each block within the blocks list contains the following parameters:

• id (integer): Represents the block's ID.
• block_center (list): Specifies the center of the block.
• block_radius (float): Represents the radius of the circular block.

Assets

We provide three object models located in the ./obj directory. It includes cube.stl, Spot and spot100.obj. The spot100.obj is generated by dividing from Spot model. You can simulate your own objects by adding your *.obj or *.stl model and modifying the JSON configuration file.

For 2D simulations, you can construct your 2D mesh by overwriting the construct_2d_mesh function in object.py file.

Reference

• FEM Simulation of 3D Deformable Solids
• An Introduction to the Conjugate Gradient Method Without the Agonizing Pain
• Keenan's 3D Model Repository