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

Published

July 25, 2024

Author(s)

Ian Bell

Abstract

The use of quadtrees to tile rectangular domains is a well accepted numerical technique. If each leaf of the quadtree is in turn a two-dimensional bi-variate approximation, a representation of a function $z(x,y)$ can be constructed that covers the entire rectangular domain. Through the use of adaptive subdivision, the function can be practically represented to any desired level of accuracy. Bi-variate Chebyshev expansions are used in each leaf due to their favorable convergence characteristics and availability in existing numerical libraries. 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 multi-fluid 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 $10^8}$ which is practically equal to the iteration convergence criterion

Citation

Industrial and Engineering Chemistry Research

Volume

63

Pub Type

Journals

Download Paper

https://doi.org/10.1021/acs.iecr.4c01631

Local Download

Citation

Bell, I. (2024), Efficient Approximation with Space Filling Quadtrees: Application to Phase Equilibria in Binary Mixtures, Industrial and Engineering Chemistry Research, [online], https://doi.org/10.1021/acs.iecr.4c01631, https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=957529 (Accessed October 13, 2025)