Efficient Approximation with Space Filling Quadtrees: Application to Phase Equilibria in Binary Mixtures

A relatively succinct set of property calculations can
be used
to construct efficient (in memory and speed) numerical approximations
of that function in a rectangular domain. Through the use of adaptive
subdivision with quadtrees and bi-Chebyshev expansions in each leaf,
the function can be practically represented to the order of the noise
in the function to be approximated. A further benefit is that evaluation
of the approximation is noniterative and thus cannot fail to converge
(a relatively common problem in thermophysical property libraries,
especially for mixtures). Evaluation of the approximation function
requires only a few bisection steps to identify the leaf of interest
such that evaluation of the approximation data structure takes less
than a microsecond. The technique is demonstrated by application to
the vapor–liquid-equilibria evaluated with two different models
(COSMO-SAC activity coefficient model and multifluid model). For the
more expensive COSMO-SAC case, the approximation function is more
than 2000 times faster to evaluate, and deviations in pressure are
less than a part in 108 which is practically equal to the
iteration convergence criterion.