P11

Particle Collision Detection --- SDF

Signed Distance Function

A signed distance function \(\phi (\mathbf{x} )\) defines the distance from \(\mathbf{x}\) to a surface with a sign. The sign indicates on which side \(\mathbf{x}\) is located.

P12

Signed Distance Function Examples

✅ 圆柱SDF基于勾股定理,\(\sqrt{\cdot }\) 内第一项为斜边长, 第二项为底边长,得出点到中轴的距离。

P13

Intersection of Signed Distance Functions

If \(\phi _0(\mathbf{x} )<0\) and \(\phi_1(\mathbf{x} )<0\) and \(\phi_2(\mathbf{x} )<0\)
then inside
\(\quad \phi (\mathbf{x} )\)=max \(⁡(\phi_0(\mathbf{x}),\phi_1(\mathbf{x}),\phi_2(\mathbf{x}))\)
Else outside
\(\quad \phi (\mathbf{x})=?\)

P14

Union of Signed Distance Functions

✅ 有时候此公式不成立,例如图中\(\mathbf{x}\) 点

Intuitively, we can consider collision detection with the union of two objects as collision detection with two separate objects.

P15

Particle Collision Response ——Penalty Method

Quadratic Penalty Method

A penalty method applies a penalty force in the next update. When the penalty potential is quadratic, the force is linear.

✅ 力的大小与距离有关,方向为normal
✅ 存在的问题:只有\(\mathbf{x}\) 进入 mesh 内部了,才会有力,但此时穿透的 artifacts 已经产生了。解决方法:使用buffor

P16

Quadratic Penalty Method with a Buffer

A buffer helps lessen the penetration issue. But it cannot strictly prevent penetration, no matter how large \(k\) is.


✅ 存在的问题:
如果 \(k\) 太小,快速的碰撞仍会产生 artifacts
如果 \(k\) 太大,碰撞的反弹过于强烈(overshooting)
解决方法:不用常数 \(k\) ,而是 \(k\) 与距离相关

P17

Log-Barrier Penalty Method

A log-barrier penalty potential ensures that the force can be large enough. But it assumes \(\phi (\mathbf{x} ) < 0\) will never happen!!! To achieve that, it needs to adjust \(\Delta t\).

✅ 用倒数关系代替线性关系。
✅ 存在的问题:
1.当\(\mathbf{x}\) 靠近物体表面时,仍然会 overshooting
2.\(\mathbf{x}\) 穿透表面后,会越陷越深。
3.本算法要求保证穿透永远不会发生,因此要仔细调节 \(\Delta t\).

P18

A Short Summary of Penalty Methods

  • The use of step size adjustment is a must.

    • To avoid overshooting.
    • To avoid penetration in log-barrier methods.
  • Log-barrier method can be limited within a buffer as well.

    • Li et al. 2020. Incremental Potential Contact: Intersection- and Inversion-free Large Deformation Dynamics. TOG.
    • Wu et al. 2020. A Safe and Fast Repulsion Method for GPU-based Cloth Self Collisions. TOG.
  • Frictional contacts are difficult to handle.

✅ 缺点:(1)难以模拟摩擦。(2)碰撞->施加力->调整,因此效果是滞后的。 优点:易实现
✅ 隐式积分比显式积分好,因为显式不稳定。

P19

Particle Collision Response —— Impulse Method

An impulse method assumes that collision changes the position and the velocity all of sudden.

✅ Penalty 方法是碰撞 → 力 → 下一时刻的速度和位置,效果滞后。
Impulse方法碰撞时立即更新速度和位置

✅ lmpulse 省去了力这一步,直接更新刚体状态。方法要求已经有一个比较好的\(\phi (x)\)
✅ 更新方法:N方向。更新距离:穿入的距离。

P20
Changing the position is not enough, we must change the velocity as well.

✅ \(\mathbf{v}\cdot \mathbf{N}\ge 0\):当前速度想要让物体越陷越深, 这种情况下才需要更新速度
把\(\mathbf{v}\)分解为\(\mathbf{v_T}\)(切线方向的速度)和\(\mathbf{v_N})\)(法线方向的速度).
✅ \(\mathbf{v_T}\)方向速度反弹, \(\mu _\mathbf{N}\) 为反弹系数。\(\mathbf{v_N}\)方向不变或由于摩擦再衰减
✅ a的约束:(1)越小越好,尽量把速度衰减掉(2)满足库仑定律(切方向的速度改变不应大于法线方向的速度改变)(3)切方向速度不能反转,即a不能为负
✅ 优点:可以精确控制摩擦力和反弹位置。缺点:计算比 Penalty 复杂
✅ 刚体常见于 Impulse; 弹性体常见于Pealty.

P21


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