P39

Shape Matching

✅ Shape Matching 跳过了。

P40

Shape Matching

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

Shape Matching

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

P42

Shape Matching

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.

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

P43

A Summary For the Day

• A mass-spring system

  • Planar springs against stretching/compression $\quad$- replaceable by co-rotational model
  • Bending springs $\quad$- replaceable by dihedral or quadratic bending
  • Regardless of the models, as long as we have $E (\mathbf{x})$, we can calculate force $\mathbf{f} (\mathbf{x} )=−∇ \mathbf{E} (\mathbf{x})$ and Hessian $\mathbf{H} (\mathbf{x} )=∂E^2(\mathbf{x} )/∂\mathbf{x} ^2$. Forces and Hessians are stackable.

• Two integration approaches

  • Explicit integration, just need force. Instability
  • Implicit integration, as a nonlinear optimization problem
  • One way is to use Newton’s method, which solves a linear system in every iteration:

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

• There are a variety of linear solvers (beyond the scope of this class).
• Some simulators choose to solve only one Newton iteration, i.e., one linear system per time step.

本文出自CaterpillarStudyGroup，转载请注明出处。

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