Position Based Dynamics (PBD)

算法过程

Position based dynamics (PBD) is based on the projection function.

• The stiffness behavior, i.e., how tightly constraints are enforced, is subject to non-physical factors.

  • The number of iterations
  • The mesh resolution

• The velocity update following projection is important to dynamic effects.

• This method is applicable to other constraints as well, including triangle constraints, volume constraints, and collision constraints.

  • To implement these constraints, simply define their projection functions.

A PBD Simulator
//Do Simulation, update \(\mathbf{x}\) and \(\mathbf{v}\)
$$ \mathbf{v}\longleftarrow\dots $$

$$ \mathbf{x}\longleftarrow\dots
$$

✅ 第一步：不考虑约束，基于粒子运动方法更新 \(\mathbf{v}\) 和 \(\mathbf{x}\)；

//Now PBD starts.

$$ \mathbf{x} ^{\mathbf{new} } \longleftarrow \mathrm{Projection} (\mathbf{x} ) $$

$$ \mathbf{v}\longleftarrow \mathbf{v} +(\mathbf{x} ^{\mathbf{new} }−\mathbf{x})/∆t $$

$$ \mathbf{x}\longleftarrow \mathbf{x} ^{\mathbf{new}} $$

✅ 第二步：基于约束和投影函数更新 \(\mathbf{v}\) 和 \(t\).
✅第二步中速度更新很重要，会影响dynamic模拟的效果。
✅\(\mathbf{v}\)的更新不是直接覆盖，而是叠加。

P12

Pros and Cons of PBD

• Pros
  • Parallelable on GPUs (PhysX)
  • Easy to implement
  • Fast in low resolutions
  • Generic, can handle other coupling and constraints, including fluids

✅ 一般来说，少于1000个点时能实时，多于1000个点时效率明显下降
✅ PBD 适用于低分辨率场景、常见的低精度实时模拟的套路。
❗ 模拟真正的时间开销不在计算 (虽然有很多计算公式) 而是在内存的访问上。
PBD 的优点是内存访问少、因为它没有太多物理变量。
因此，对追求效率的场景，主要优化内存访问而不是计算。

• Cons
  • Not physically correct
  • Low performance in high resolutions
    • Hierarchical approaches (can cause oscillation and other issues…)
    • Acceleration approaches, like Chebyshev

✅ 问：PBD 非物理方法，怎么体现出弹性效果？
答：迭代数多则弹性差、网格顶点少则弹性差。
✅ 弹性表现受网格数量影响(难以控制)、没有所谓的精确解，哪怕迭代数足够多、迭代数过多会导致locking issue.

P13

After-Class Reading

Muller. 2008. Hierarchical Position Based Dynamics. VRIPHYS.

✅ NVIDIA的很多物理引擎都是基于PBD的

本文出自CaterpillarStudyGroup，转载请注明出处。

https://caterpillarstudygroup.github.io/GAMES103_mdbook/