GAME201 HW1

Implemented explicit and implicit euler of mass spring.

Explicit euler, implemented

Forward euler
RK2
RK3
RK4

Implicit euler, implemented

Backward euler with jacobi iteration (#iteration=20)

More details:

Code was based on mass_spring_explicit.py example.
Moved "Collide with ground" to after "Compute new position" because my explicit code updates x and v at the same time (I didn't know whether this was acceptable or not)
Didn't know how to compute gradient in implicit euler. Used auto-grad in taichi to compute the gradient.
Settings:
- #particles = 10
- damping = 20.0
- dt = 1e-3

Comparsion

\	Forward Euler	RK2	RK3	RK4	Implicit euler
latency per step *	~0.4ms	~0.5ms	~0.6ms	~0.7ms	~5m (~3m w/o auto-grad computing time)
energy stableness **	~0.92	~0.92	~0.92	~0.92	~0.98
energy stableness (damping=0.0) *******	unstable	~1.0	~1.0	~1.0	~0.99 - ~1.0
max spring stiffness	~7,500	~45,000	~300,000	~700,000	~300,000 (similar value for #iteration=50)
max dt (spring stiffness=1000)	~2e-3	~8e-3	~0.02	~0.03	~0.15

* environment (windows 10, ram=16G, cpu=AMD Ryzen 1700X, gpu=Nvidia GTX 1700Ti)

** energy (mgh + mv^2 / 2 + kΔx^2 / 2) stableness = moving average rate of (energy at step t / energy at step t-100) after the rate becomes stable and before the springs hit the ground (the rate is ~1.0 after the springs hit the ground)

*** Record rate after the rate becomes stable.

Observation

When damping exists, energy stableness of implicit euler is better; otherwise, all methods are great.
When damping exists, all simulation looks similar; otherwise, all looks different (please check the following GIFs)
RK3 / RK4 / implicit euler provide good max spring stiffness.
Implicit euler provides much better max dt.

GIFs (damping = 20.0)