P39

Shape Matching

先让每个粒子独立仿真，也不需要考虑弹簧力。
将每个面片的变形程度构建为能量，通过优化能量的方式使 Mesh 减少形变，所有面片以统一的方式进行优化，因此是一种全局优化方法。

所有方法最后都归结为一个分式：

$$ \psi(x)=k||x-y||_m^2+E(x) $$

关键是如何定义，以及非线性方程如何求解

P40

量化形变

The basic idea is to define a quadratic energy based on the rotated reference element. To do so, we split transformation into deformation + rotation.

P41

P42

计算能量

We can then define the quadratic energy as:

$$ E (\mathbf{x} )=\frac{1}{2}||\mathbf{F−R} ||^2 $$

($\mathbf{R}$ is the rotation inside of $\mathbf{F}$. This energy tries to penalize the existence of $\mathbf{S}$).

Assuming that $\mathbf{R}$ is constant, this $E(\mathbf{x})$ becomes a quadratic function. We can then derive the force and the Hessian.

$$ E(\mathbf{x} ) =\frac{1}{2} ||\begin{bmatrix} \mathbf{x} _1-\mathbf{x} _0 &\mathbf{x} _2-\mathbf{x} _0 \end{bmatrix}\begin{bmatrix} \mathbf{r} _1-\mathbf{r} _0 &\mathbf{r} _2-\mathbf{r} _0 \end{bmatrix}^{−1}−\mathbf{R}||^2 $$

P24

Shape Matching

这是用 $PD$ 来优化优化问题的另一个例子。原 $PD$ 算法用于长度保持约束，而这里把它用于形状保持约束。

Shape matching is also projective dynamics, if we view rotation as projection:

The 2D Space	The 3D Space

Assuming that $\mathbf{{\color{Orange} R} }$ is constant,
$$ \begin{matrix} \mathbf{f} _0=−\nabla_0E(\mathbf{x} )\\ \mathbf{f} _1=−\nabla_1E(\mathbf{x} ) \\ \mathbf{f} _2=−\nabla_2E(\mathbf{x} )\\ \mathbf{H} =\frac{∂E^2(\mathbf{x} )}{∂x^2} \quad \text{is a constant !}
\end{matrix} $$

P25

Simulation by Projective Dynamics

According to implicit integration and Newton’s method, a projective dynamics simulator looks as follows, with matrix $\mathbf{A} =\frac{1}{∆t^2}\mathbf{M+}\mathbf{H} $ being constant.
We can use a direct solver with only one factorization of A.

Pipeline

Initialize $\mathbf{x} ^{(0)}$, often as$ \mathbf{x} ^{[0]} $or $\mathbf{x} ^{[0]} +∆t\mathbf{v} ^{[0]} $

For $k=0\dots K$
$\quad$ Recalculate projection

$\quad$ Solve $(\frac{1}{∆t^2}\mathbf{M} +\mathbf{H} )∆\mathbf{x} =−\frac{1}{∆t^2}\mathbf{M} (\mathbf{x} ^{(k)}−\mathbf{x} ^{[0]}−∆t\mathbf{v} ^{[0]})+\mathbf{f} (\mathbf{x} ^{(k)})$

$\quad$ $\mathbf{x} ^{(k+1)}\longleftarrow \mathbf{x} ^{(k)}+∆\mathbf{x} $

$\quad$ If $||∆\mathbf{x}||$ is small $\quad$ then break

$\mathbf{x} ^{[1]}\longleftarrow \mathbf{x} ^{(k+1)}$

$\mathbf{v} ^{[1]}\longleftarrow (\mathbf{x} ^{[1]}-\mathbf{x} ^{[0]})/∆t$

P27

Pros and Cons of Projective Dynamics

Pros

By building constraints into energy, the simulation now has a theoretical solution with physical meaning.
Fast on CPUs with a direct solver. No more factorization!

✅ Fast on CPU,因为它只作一次$\mathbf{LU}$分解。

Fast convergence in the first few iterations.

Cons

Slow on GPUs. (GPUs don’t support direct solver wells.)

✅ Slow on GPU，因为$\mathbf{LU}$分解不适用于 $\mathbf{GPU}$

Slow convergence over time, as it fails to consider Hessian caused by projection.
- Still suffering from high stiffness
Cannot easily handle constraint changes.
- Contacts
- Remeshing due to fracture, etc.

✅ constraint changes: 网格关系改变导至弹簧结构改变，原来的$\mathbf{H}$将不再适用。

本文出自CaterpillarStudyGroup，转载请注明出处。

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