Finite Element Simulation

Developed deformable solid objects simulation based on tetrahedral meshes using FEM.

Code Logic

• Extracting the surface mesh

  For each face, sort the vertex indices and store to the map <index, index, index> to <count>. Loop through all the tets to build the map and surface meshes is faces with count == 1.

• Computing and applying internal forces

  Calculate F and stress for each tet and loop through particles to update internal forces based on f = F * stress * area * normal. Detailed calculation of F and stress is shown on utils.cpp.

• Collision resolution

  When a particle hits the collider, apply a penalty force to the tet in the direction of the normal of the collider. The magnitude of the penalty force is calculated by f = k * |v|. Any particle that is inside the collider will be projected out later.

• Explicit integration method

  Store the original state and update the position and velocity of particles on midpoint. Then calculate one step dx and dv on the midpoint state and apply them to the original state.

Extra Features

• Runge-Kutta 4 Integrator

  Store the original state. Take one step from original state and store dx, dv to k1. Take half step from original state and store dx, dv to k2. Take half step from k2 and store dx, dv to k3. Take one step from k3 and store dx, dv to k4. Update from original state with delta = 1 / 6 * (k1 + 2k2 + 2k3 + k4).

• Adaptive time stepping

  Take one step from original state to get state xa. Take two half step from original state to get state xb. Calculate the error based on xa and xb. Update the time step t to t' = sqrt(max_error / error) * t.

Demo Video

• Single tetrahedron falling on the floor (./inis/tet.ini $\rightarrow$ ./demo-video/tet.mp4)

• Cube falling on the floor (./inis/cube.ini $\rightarrow$ ./demo-video/cube.mp4)

• Ellipsoid falling on the floor with fixed sphere (./inis/ellipsoid.ini $\rightarrow$ ./demo-video/ellipsoid.mp4)

• Higher-order explicit integrator

  • With Runge-Kutta 4 Integrator (./inis/rk4_on.ini $\rightarrow$ ./demo-video/rk4_on.mp4)

  • Without Runge-Kutta 4 Integrator (./inis/rk4_off.ini $\rightarrow$ ./demo-video/rk4_off.mp4)

• Adaptive time stepping

  • With Adaptive time stepping (./inis/adapt_on.ini $\rightarrow$ ./demo-video/adapt_on.mp4)

  • Without Adaptive time stepping (./inis/adapt_off.ini $\rightarrow$ ./demo-video/adapt_off.mp4)