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Strain Based Dynamics

Matthias Müller, Nuttapong Chentanez, Tae-Yong Kim, Miles Macklin

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PBD extension constraining Green-St Venant strain tensor entries directly, enabling anisotropic cloth behavior without polar decomposition overhead.

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Abstract

This paper introduces a new set of constraints within the Position Based Dynamics framework that control strain in directions independent of the simulation mesh edges. Instead of constraining distances between points, the method constrains the entries of the Green St Venant strain tensor, so varying the per-coefficient stiffness produces anisotropic behavior without the polar decomposition required by most strain limiting approaches. The authors propose modified diagonal constraints solvable in a single step and modified off-diagonal constraints that decouple stretch from shear resistance, and they add separate volume and area conservation constraints. Because the constraints are formulated within PBD, they can drive the actual simulation of deformable objects such as cloth and tetrahedral solids rather than only limiting overstretched material.

How to read this

Category: Method: a Position Based Dynamics extension for strain-based deformable simulation
Contributions: Constraints on the entries of the Green St Venant strain tensor rather than on mesh edge distances, giving anisotropy without polar decomposition
Modified diagonal constraints solvable in a single step and off-diagonal constraints that decouple stretch from shear
Separate volume and area conservation constraints, with the formulation driving the actual simulation rather than only limiting overstretch
Context: A direct extension of 'Position Based Dynamics', recasting elastic behavior as strain-tensor constraints within the PBD constraint-projection framework.Builds on: Position Based Dynamics
Correctness: Works within PBD, so stiffness behavior is iteration- and timestep-dependent as is typical for the method, and results are demonstrated on cloth and tetrahedral solids; readers should remember PBD strain control is geared to plausible, controllable behavior rather than physically calibrated material response.
Clarity: Accessible if you know PBD; a first pass conveys the strain-constraint idea, a second pass gives the per-constraint projection formulas.
How to read it: Read with the PBD paper at hand; focus first on what each strain-tensor constraint controls, then do a second pass on the diagonal and off-diagonal constraint derivations to implement the solver step.

Builds on

Position Based Dynamics 2007

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Related work

Nonlinear Cloth Simulation with Isogeometric Analysis 2024 / CASA
Multi-Resolution Isotropic Strain Limiting 2010 / SIGGRAPH Asia
Projective Dynamics: Fusing Constraint Projections for Fast Simulation 2014 / SIGGRAPH
Codimensional Incremental Potential Contact 2021 / SIGGRAPH

Keywords

position based dynamics strain based dynamics green strain tensor strain limiting deformation gradient cloth simulation anisotropic material tetrahedral mesh shear and stretch decoupling volume conservation finite element method mass spring system polar decomposition skin sliding linear blend skinning bending constraint soft body simulation

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