PBD algorithm

Setup

• Mass points :

  • Mass 𝒎𝒊, position 𝒙𝒊, velocity 𝒗𝒊

• Constraints :

  • e.g. spring(distance) constraint

• Just like mass spring systems

• Main difference time stepping method

Constraints & Constraint Projection

• Operate on “proposed solution” : 𝒑𝟏,…,𝒑𝒏

  • Distance constraint (spring) : 𝑪𝒑𝒊,𝒑𝒋=𝒑𝒊−𝒑𝒋−𝒅

  • The most primitive of geometric constraints, distance based constraints restore mesh edges that extend or compress relative to their rest length

• Projection

  • Task : update state 𝒑𝟏,…,𝒑𝒏so that the constraints are approximately satisfied

  • PBD uses local linearization : one constraint at a time

  • PBD use only first derivatives

Constraint Projection

-Main idea of PBD

• Project constraints one by one

• One descent step per constraint

• Why this works better?

  • 𝒑𝒊are immediately updated

  • Next linearization works with the updated state

Constraint Projection Math

• Taylor expansion :

  • 𝑪𝒑+Δ𝒑≈𝑪𝒑+𝜵𝒑𝑪𝒑∙Δ𝒑=𝟎

• Step :

  • Δ𝒑=𝝀𝜵𝒑𝑪𝒑

Putting together :

• Δ𝒑=−𝑪(𝒑)/|𝜵𝒑𝑪(𝒑)|^𝟐*𝜵𝒑𝑪𝒑

Other Constraint : Bending

• Important for cloth and thin shells

• How to measure bending?

  • Simple way : cross spring

  • Drawback : interferes with in plane stretching

Pressure Constraints

• Constraint on volume of closed mesh

  • e.g. Cloth balloons

  • 

    • 𝒗𝟎: original volume

    • 𝒌𝒑𝒓𝒆𝒔𝒔𝒖𝒓𝒆: over pressure factor

Collision Constraints

• Assume : collisions already detected

  • Simple objects : easy

• Similar as penalty forces (but constraints)