🧩 Non-rigid Dynamics#

This page gives a compact overview of the physical models implemented by Genesis’ continuum and discrete solvers. The emphasis is on which equations are being solved and how, rather than on the Python API. For coupling theory see the dedicated Solvers & Coupling chapter.

1. Eulerian Stable-Fluid Solver (SFSolver)#

Purpose. Fast smoke / gas simulation on a fixed grid.

Governing equations – incompressible Navier–Stokes.

Algorithm – Jos Stam’s Stable Fluids:

  1. Advection – velocities are back-traced with third-order RK and interpolated (backtrace + trilerp). Numerically unconditionally stable.

  2. External impulses – jet sources inject momentum after advection.

  3. Viscosity / decay – optional exponential damping term.

  4. Pressure projection – solve Poisson by Jacobi iteration (pressure_jacobi).

  5. Boundary conditions – zero-normal velocity enforced by mirroring components at solid faces.

Because all steps are explicit or diagonally implicit the method is extremely robust at large time-steps and suitable for real-time effects.

2. Material Point Method Solver (MPMSolver)#

Purpose. Unified simulation of solids, liquids and granular media using particles + background grid.

Core idea. The continuum momentum equation is evaluated on an Eulerian grid while material history (deformation gradient, plastic strain, etc.) is stored on Lagrangian particles.

2.1 Update sequence (APIC / CPIC variant)#

Phase

Description

P2G

Transfer mass and momentum to neighbour grid nodes with B-spline weights; add stress contribution.

Grid solve

Divide by mass to obtain velocities, apply gravity & boundary collisions.

G2P

Interpolate grid velocity back, update affine matrix and position.

Polar-SVD

Decompose deformation graident; material law returns new deformation gradient.

2.2 Constitutive models#

Genesis ships several analytic stress functions:

  • Neo-Hookean elastic (chalk/snow)

  • Von Mises capped plasticity (snow-plastic)

  • Weakly compressible liquid (WC fluid)

  • Anisotropic muscle adding active fibre stress

3. Finite Element Method Solver (FEMSolver)#

Purpose. High-quality deformable solids with tetrahedral meshes; optional implicit integration for stiff materials.

3.1 Energy formulation#

Total potential energy

\[ \Pi(\mathbf x) = \sum_{e} V_{e}\,\psi(\mathbf F_e) - \sum_{i} m_{i}\,\mathbf g\!\cdot\!\mathbf x_i. \]

The first variation yields the internal force; the second variation gives the element stiffness.

3.2 Implicit backward Euler#

Given current state \((\mathbf x^n, \mathbf v^n)\) solve for \(\mathbf x^{n+1}\) by Newton–Raphson:

\[ \mathbf r(\mathbf x) = \frac{m}{\Delta t^{2}}(\mathbf x - \hat{\mathbf x}) + \frac{\partial \Pi}{\partial \mathbf x} = 0,\]

where \(\hat{\mathbf x} = \mathbf x^{n} + \Delta t\,\mathbf v^{n}\) is the inertial prediction.

Each Newton step solves \(\mathbf H\,\delta \mathbf x = -\mathbf r\) with PCG; \(\mathbf H\) is the consistent stiffness + mass matrix. A block-Jacobi inverse of per-vertex 3×3 blocks is used as preconditioner. Line-search (Armijo back-tracking) guarantees energy decrease.

4. Position-Based Dynamics Solver (PBDSolver)#

Purpose. Real-time cloth, elastic rods, XPBD fluids and particle crowds.

4.1 XPBD integration cycle#

  1. Predict – explicit Euler: \(\mathbf v^{*}\!=\!\mathbf v + \Delta t\,\mathbf f/m\) and \(\mathbf x^{*}\!=\!\mathbf x + \Delta t\,\mathbf v^{*}\).

  2. Project constraints – iterate over edges, tetrahedra, SPH density etc. For each constraint \(C(\mathbf x)\!=\!0\) solve for Lagrange multiplier λ

    \[ \Delta\mathbf x = -\frac{C + \alpha\,\lambda^{old}}{\sum w_i\,|\nabla\!C_i|^{2}+\alpha}\,\nabla\!C, \quad \alpha = \frac{\text{compliance}}{\Delta t^{2}}. \]

  3. Update velocities – \(\mathbf v = (\mathbf x^{new}-\mathbf x^{old})/\Delta t\).

4.2 Supported constraints#

  • Stretch / bending (cloth)

  • Volume preservation (XPBD tetrahedra)

  • Incompressible SPH density & viscosity constraints (fluid-PBD)

  • Collision & friction via positional correction + Coulomb model.

5. Smoothed Particle Hydrodynamics Solver (SPHSolver)#

Purpose. Particle-based fluids with either WCSPH or DFSPH pressure solvers.

5.1 Kernels#

Cubic spline kernel \(W(r,h)\) and gradient \(\nabla W\) with support radius \(h=\) _support_radius.

5.2 Weakly Compressible SPH (WCSPH)#

  • Equation of state: \(p_i = k\bigl[(\rho_i/\rho_0)^{\gamma}-1\bigr]\).

  • Momentum equation:

    \[ \frac{d\mathbf v_i}{dt} = -\sum_j m_j \left( \frac{p_i}{\rho_i^2} + \frac{p_j}{\rho_j^2} \right) \nabla W_{ij} + \mathbf g + \mathbf f_{visc} + \mathbf f_{surf}. \]

5.3 Divergence-free SPH (DFSPH)#

  • Splits solve into divergence pass (enforce \(\nabla\!\cdot\mathbf v = 0\)) and density pass (enforce \(\rho\!=\!\rho_0\)).

  • Both passes iteratively compute per-particle pressure coefficient κ with Jacobi iterations using the DFSPh factor field.

  • Ensures incompressibility with bigger time-steps than WCSPH.

References#

  • Stam, J. “Stable Fluids”, SIGGRAPH 1999.

  • Zhu, Y.⁠ & Bridson, R. “Animating Sand as a Fluid”, SIGGRAPH 2005.

  • Gao, T. et al. “Robust Simulation of Deformable Solids with Implicit FEM”, SIGGRAPH 2015.

  • Macklin, M. et al. “Position Based Fluids”, SIGGRAPH 2013.

  • Bender, J. et al. “Position Based Dynamics”, 2014.

  • Bavo et al. “Divergence-Free SPH”, Eurographics 2015.