Lecture 3: Simulation Loop

28 Aug 2024

Notes Physically Based Modelling Lecture 3: Simulation Loop

§ The Loop

The basic simulation loop that we'll use in this class is to integrate, check for collisions, determine and resolve them, repeat.

Simulation Loop

Set initial conditions
$t$ $\leftarrow$ 0
while $t < t_{\mathrm{max}}$ $t < t_{max}$ do
- $T$ $\leftarrow$ $\min(h, T_{\mathrm{disp}} - t)$
- $\mathbf{a}$ $\leftarrow$ $\frac{1}{m} \mathbf{F}_{\mathrm{total}}$
- $\mathbf{x}_{\mathrm{new}}, \mathbf{v}_{\mathrm{new}}$ $\leftarrow$ Integrate( $\mathbf{x}, \mathbf{v}, T$ )
- if DetectCollision( $\mathbf{x}, \mathbf{v}, \mathbf{x}_{\mathrm{new}}, \mathbf{v}_{\mathrm{new}}$ $x, v, x_{new}, v_{new}$ ) then
  - $f$ $\leftarrow$ fractional timestep of collision
  - $\mathbf{x}_{c}, \mathbf{v}_{c}$ $\leftarrow$ Integrate( $\mathbf{x}, \mathbf{v}, T$ )
  - $\mathbf{v}$ $\leftarrow$ CollisionResponse( $\mathbf{x}_{c}, \mathbf{v}_{c}$ )
  - $\mathbf{x}$ $\leftarrow$ $\mathbf{x}_c$
- else
  - $\mathbf{x}$ $\leftarrow$ $\mathbf{x}_{\mathrm{new}}$
- $t$ $\leftarrow$ $t + T$
- if $t > T_{\mathrm{disp}}$ $t > T_{disp}$ then
  - render and output frame
  - update $T_{\mathrm{disp}}$

§ Resting Situations

We want to stop the simulation for an object whenever it comes to rest; this saves on computational resources, avoids numerical issues, and jitter effects.

For an object to come to rest, it must satisfy all of the following

$\|\mathbf{v}\| \leq \delta_1$ ; the speed of the object is less than some small threshold
There exists a surface $s$ such that $\mathrm{dist}(s, \mathbf{x}) \leq \delta_2$ ; i.e. it touches a some surface
Acceleration must be towards the touching surface, $\mathbf{a}\cdot\hat{\mathbf{n}} < 0.$
Friction must overcome the tangential acceleration, $\|\mathbf{a}_\parallel\| \leq \mu\|\mathbf{a}_\perp\|$

§ Numerical Error

Roundoff error grows and propagates as we chain computations.

We won't get into the specifics in this course, but some guidelines are to

use tolerances; do not check if $a \stackrel{?}{=} b$ but rather $\|a - b\| \stackrel{?}{\leq} \varepsilon,$ and to
not assume transitivity; e.g. equality $(A = B) \wedge (B = C) \;\not\!\!\!\implies A = C.$