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/