56 6. SIMULATION AS AN OPTIMIZATION PROBLEM
found by solving the following two linear systems sequentially
Kz D b
K
T
x D z:
(6.22)
is can be computed very efficiently since K is a lower triangular matrix that can be
precomputed during the initialization of the simulation.
6.7 CONCLUSION
In this chapter, we looked at an alternative approach to solving the time integration for cloth
simulation. e integration is reformulated as an optimization problem that can be solved using
a two-step approach. e optimization method is called local-global alternation or block coor-
dinate descent. It provides very efficient results because the left-hand side matrix of the linear
system can be precomputed and pre-factorized into a very efficient formulation, as long as the
particle connectivity doesn’t change.
So how does it compare to Newton’s method? Using this method, it will be very fast to
compute a single iteration of local-global alternation. is allows us to perform many iterations
at the same computational cost of a single Newton iteration. is is particularly handy if you
have a limited time budget like in video games that demand a specific frame rate.
After a few iterations, there’s a cutoff point where Newton’s method performs much better
than this method. is makes it clear that if you’re going for accuracy, Newton’s method is the
way to go. If, however, you have a limited computation time available to advance the simulation
to the next time step, this method might prove to be of great value to you.
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