Physical Dynamics and Constraints › Deformable Bodies

Not everything is rigid. Continuum-based deformable models discretize a body into many degrees of freedom, together with velocities and material state; other families begin directly from discrete particles and constraints. At finite resolution, the element type, mesh, integration rule, and parameter calibration can materially affect behavior. Consistent convergent discretizations should approach the same continuum response under refinement, but two coarse or differently calibrated cloth models can behave unlike each other.

The main discretization families, each with a distinct modeling tradeoff, are:

• Mass–spring systems: Represent the body as particles connected by elastic springs, often with damping. They are easy to implement, but the material behavior depends on the mesh, and capturing volume preservation or buckling requires care. Two different meshes of the same object can behave differently, which is a symptom that the model is describing the discretization rather than the material.
• Finite element methods (FEM): Discretize a continuum-mechanics model into elements and integrate its constitutive response. FEM supports linear elasticity, hyperelasticity, and plasticity; stability and accuracy depend on the formulation, mesh, elements, and solver. Parameters such as Young's modulus, Poisson's ratio, and yield stress can then be estimated from experiments when they are identifiable.
• Position-based dynamics (PBD): Solve constraint projections directly on particle positions rather than integrating forces. Extremely stable and fast, which made it the standard in interactive graphics (Müller et al., 2007), but its convergence behavior means results are iteration- and timestep-dependent. Extended Position Based Dynamics (XPBD) restores physical consistency by making compliance explicit, decoupling stiffness from the timestep and iteration count.
• Material point method (MPM): Hybridize material-carrying particles with a background grid. Graphics research demonstrated snow simulation for feature films, and related particle-grid methods have been applied to other granular and fluid-like materials. The representation accommodates large deformation and topology change without a permanently connected mesh, although fracture and material behavior still depend on the constitutive model and discretization.

Why should a world-model builder care about these simulation methods? High-fidelity deformable simulation can be costly, so an offline simulator can serve as a teacher for a faster learned surrogate. The teacher still matters: a surrogate trained on PBD output inherits its finite-timestep and iteration behavior, and a planner may optimize against those artifacts. Deformable simulation also supplies a useful constraint language. Equality constraints such as a cloth edge's rest length or a volume residual are held at $C_i(x)=0$. Unilateral contact is different: a signed gap is kept at $\phi(x)\geq0$, with equality only at contact.