Soft solvers#

Beyond the rigid solver, Genesis World ships five solvers for continuum and particle materials. Assigning a material to an entity selects both the solver that advances it and the constitutive model it obeys, the rule that turns deformation into stress. The simulator activates a solver at build time only if it holds at least one entity.

To pick a solver and run one, see Beyond rigid bodies. For how forces cross material boundaries, see Coupling.

Solvers#

Solver

Models

Representation

MPM (Material Point Method)

Elastic solids, plastics, sand, snow, liquids

Particles carry state, a background grid resolves forces

FEM (Finite Element Method)

Stiff elastic solids, volumetric muscles, thin shells

Tetrahedral mesh

PBD (Position-Based Dynamics)

Cloth, ropes, topology-preserving deformables

Particles linked by constraints

SPH (Smoothed-Particle Hydrodynamics)

Free-surface liquids

Particles

SF (Stable Fluids)

Smoke and gas

Fixed Eulerian grid

A kinematic solver for scripted motion and a tool solver for driven manipulators round out the set, and a material selects them as it does the five above: gs.materials.Kinematic and gs.materials.Tool. The tool solver drives the soft solvers through one-way coupling, while a kinematic entity is rendered only and takes no part in physics.

Deformation and stress#

Every continuum material tracks the deformation gradient \(\mathbf F\), the local map from an undeformed neighborhood to its deformed shape. Its determinant \(J = \det \mathbf F\) is the local volume ratio: \(J = 1\) preserves volume, \(J < 1\) is compression, \(J > 1\) expansion. A constitutive model turns \(\mathbf F\) into a stress, usually through an elastic strain-energy density \(\psi(\mathbf F)\) whose derivative is the force.

Most solids take three parameters, from which Genesis derives the Lamé coefficients \(\mu\) (resistance to shear) and \(\lambda\) (resistance to volume change):

  • E: Young’s modulus in Pa. Larger E is a stiffer body, and a numerically stiffer system that needs smaller substeps.

  • nu: Poisson ratio. Values near 0.5 are nearly incompressible.

  • rho: density in kg/m³, or kg/m² for the 2D PBD cloth model.

\[\mu = \frac{E}{2(1+\nu)}, \qquad \lambda = \frac{E\,\nu}{(1+\nu)(1-2\nu)}.\]

Several models factor \(\mathbf F\) first. The polar decomposition \(\mathbf F = \mathbf R \mathbf S\) splits it into a rotation and a symmetric stretch; the singular value decomposition \(\mathbf F = \mathbf U \boldsymbol\Sigma \mathbf V^\top\) exposes the principal stretches on the diagonal of \(\boldsymbol\Sigma\). Plasticity models operate on those stretches directly.

Elastic solids#

An elastic material returns to its rest shape when unloaded. The elastic classes are also the base that the plastic and muscle models extend.

gs.materials.MPM.Elastic selects its stress model through model:

  • "corotation" (default): fixed-corotated, \(\psi(\mathbf F) = \mu\,\lVert \mathbf F - \mathbf R\rVert_F^2 + \tfrac{\lambda}{2}(J-1)^2\). The energy penalizes deviation from the nearest rotation, so large rotations stay well behaved.

  • "neohooken": \(\psi(\mathbf F) = \tfrac{\mu}{2}(\operatorname{tr}(\mathbf F^\top\mathbf F) - 3) - \mu\ln J + \tfrac{\lambda}{2}(\ln J)^2\). Reads \(\mathbf F\) and \(J\) directly and skips the SVD, so it costs less per particle.

gs.materials.FEM.Elastic solves elasticity on a tetrahedral mesh, defaulting to "linear":

soft = scene.add_entity(
    material=gs.materials.FEM.Elastic(E=3e5, nu=0.45, model="stable_neohookean"),
    morph=gs.morphs.Sphere(radius=0.1),
)

  • "linear": linear elasticity, the only model with a constant precomputed Hessian. Valid for small strains; large rotations produce artifacts.

  • "stable_neohookean": rest-stable Neo-Hookean. Its energy stays defined for inverted or degenerate elements, which suits large deformation and contact-rich scenes.

  • "linear_corotated": linear elasticity in a per-element rotated frame, correct under large rotation with a linear response to stretch.

gs.materials.PBD.Elastic integrates no stress. Position-Based Dynamics enforces geometric constraints on particle positions, and expresses stiffness as compliance, the inverse: stretch_compliance, bending_compliance, and volume_compliance govern edge, bending, and volume constraints. Under XPBD a constraint’s effective compliance is \(\alpha = \text{compliance}/\Delta t^2\), so 0.0 is perfectly rigid. It trades physical accuracy for speed and stability.

Plastic solids#

A plastic material keeps part of its deformation after unloading. Genesis splits \(\mathbf F\) into an elastic part that stores energy and a plastic part that does not: each step computes a trial elastic state, then a return mapping projects it onto a yield surface and moves the excess into the plastic part.

gs.materials.MPM.ElastoPlastic selects between two yield criteria through use_von_mises:

  • von Mises (default): yielding follows the deviatoric part of the Hencky strain \(\boldsymbol\varepsilon = \ln\boldsymbol\Sigma\). The material flows once \(\lVert \operatorname{dev}\boldsymbol\varepsilon\rVert\) exceeds \(\tau_Y / (2\mu)\), with von_mises_yield_stress setting \(\tau_Y\), so the body dents and holds the dent.

  • Singular-value clamping (use_von_mises=False): the principal stretches are clamped into \([\,1-\texttt{yield\_lower},\ 1+\texttt{yield\_higher}\,]\), capping elastic stretch and compression before the rest becomes permanent.

gs.materials.MPM.Sand is a Drucker-Prager model for cohesionless granular media. Its yield surface is a cone in stress space set by friction_angle in degrees: particles resist shear only under confining pressure, so sand holds an angle of repose and otherwise flows.

gs.materials.MPM.Snow specializes ElastoPlastic with singular-value clamping only, and hardens as it compacts, so it packs into a shape-holding solid.

Liquids#

Liquids sustain no shear stress at rest and resist only volume change. The three classes differ in how incompressibility is enforced.

gs.materials.MPM.Liquid is weakly compressible: each step discards the shape of \(\mathbf F\) and keeps its volumetric part \(J^{1/3}\mathbf I\), so no shear stress accumulates and pressure comes from volume change alone. viscous=True retains a deviatoric viscous term for a thicker fluid.

gs.materials.SPH.Liquid derives pressure from a Tait equation of state,

\[p_i = k\left[\left(\frac{\rho_i}{\rho_0}\right)^{n} - 1\right],\]

with \(k\) from stiffness, \(n\) from exponent, and rest density \(\rho_0\) from rho. Viscosity and surface tension are set directly, by mu and gamma.

gs.materials.PBD.Liquid enforces a per-particle density constraint positionally rather than through pressure, tuned by density_relaxation and viscosity_relaxation.

Cloth#

Cloth stretches and bends with negligible thickness.

  • gs.materials.PBD.Cloth: a constraint-based sheet with separate stretch_compliance and bending_compliance. Its rho is a surface density, so entity mass is rho times surface area.

  • gs.materials.FEM.Cloth: a thin-shell FEM material for the IPC contact backend, parameterized by thickness in meters and an optional bending_stiffness. Use it when cloth must resolve penetration-free contact against other bodies.

Muscles#

A muscle adds a controllable active stress along an embedded fiber direction \(\mathbf m\) on top of a passive elastic response, proportional to a per-step actuation signal. Contracting the fiber pulls the body into a new shape; releasing it lets the elastic part restore the rest configuration.

Both take n_groups to define independently actuated fiber groups. The control loop is covered in Soft robots.

Choosing a model#

Behavior

Material

Key parameter

Recoverable elastic solid

MPM.Elastic, FEM.Elastic, PBD.Elastic

model, or compliance for PBD

Dents and holds its shape

MPM.ElastoPlastic

use_von_mises, von_mises_yield_stress

Granular media

MPM.Sand

friction_angle

Compacting snow

MPM.Snow

yield_lower, yield_higher

Flowing liquid

MPM.Liquid, SPH.Liquid, PBD.Liquid

viscous; stiffness, mu, gamma for SPH

Actuated soft body

MPM.Muscle, FEM.Muscle

n_groups

Cloth and shells

PBD.Cloth, FEM.Cloth

compliances; thickness for FEM

References#

  • Stomakhin, A. et al. “A Material Point Method for Snow Simulation.” SIGGRAPH 2013.

  • Klár, G. et al. “Drucker-Prager Elastoplasticity for Sand Animation.” SIGGRAPH 2016.

  • Smith, B., Goldade, T., Kim, T. “Stable Neo-Hookean Flesh Simulation.” ACM TOG 2018.

  • Macklin, M., Müller, M., Chentanez, N. “XPBD: Position-Based Simulation of Compliant Constrained Dynamics.” MIG 2016.

  • Bender, J., Koschier, D. “Divergence-Free Smoothed Particle Hydrodynamics.” SCA 2015.