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C H A P T E R 5
Implicit Integration
“I think what children need is love, stability, consistency, and kindness.”
Rosie O’Donnell
Much like children, cloth simulations need stability and love.
5.1 INTRODUCTION
Artists love big time steps h because it advances the simulation more quickly. Large time steps
will decrease accuracy but that might not be a big problem, as long as the results are visually
pleasing, we’re happy. However, we also saw that when the time step is too large, the simulation
results will be unstable and completely unusable.
To obtain more stable simulation results, Baraff and Witkin [1998] had the excellent idea
of applying an implicit integration scheme to cloth simulations. is complicates the integration
a little bit since we will have to deal with second derivatives. All things considered, the effort is
well worth it. e idea is that implicit integration would enable us to obtain stable results, even
when using large time steps. is means that even though the computation of one step will be
more computationally intensive compared to explicit Euler, we can take larger steps to advance
time in the simulation more rapidly without having to worry about stability issues.
5.2 BACKWARD EULER
Let’s say we start at time t
n
. At each step, we want to compute our update in positions x D
x
nC1
x
n
and velocities v D v
nC1
v
n
using the following integration scheme:
x D h
.
v
n
C v
/
v D h
M
1
f.x
n
C x; v
n
C v/
:
(5.1)
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