Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Boolean algebra of two-dimensional continua with arbitrarily complex topology


Authors: Qinghai Zhang and Zhixuan Li
Journal: Math. Comp. 89 (2020), 2333-2364
MSC (2010): Primary 65D18, 76T99
DOI: https://doi.org/10.1090/mcom/3539
Published electronically: May 8, 2020
MathSciNet review: 4109569
Full-text PDF
View in AMS MathViewer New

Abstract | References | Similar Articles | Additional Information

Abstract: We propose a mathematical model for two-dimensional continua in order to establish a solid theoretical foundation for the study of their complex topology, large geometric deformations, and topological changes such as merging in the context of multiphase flows. Our modeling space, named the Yin space, consists of regular open semianalytic sets with bounded boundaries, and is further equipped with constructive and algebraic definitions of Boolean operations. Major distinguishing features of our model include (a) topological information of fluids such as Betti numbers can be easily extracted in constant time, (b) topological changes of fluids are captured by nonmanifold points on fluid boundaries, and (c) Boolean operations on fluids correctly handle all degenerate cases and apply to arbitrarily complex topologies, yet they are simple and efficient in that they only involve determining the relative position of a point to a Jordan curve and intersecting a number of curve segments. Finally, utilities of (a) and (c) are demonstrated by combining the Yin space with the recent cubic MARS method to track a complex fluid in a single vortex flow.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65D18, 76T99

Retrieve articles in all journals with MSC (2010): 65D18, 76T99


Additional Information

Qinghai Zhang
Affiliation: School of Mathematical Sciences, Zhejiang University, 38 Zheda Road, Hangzhou, Zhejiang Province, 310027 People’s Republic China
Email: qinghai@zju.edu.cn

Zhixuan Li
Affiliation: School of Mathematical Sciences, Zhejiang University, 38 Zheda Road, Hangzhou, Zhejiang Province, 310027 People’s Republic China

DOI: https://doi.org/10.1090/mcom/3539
Received by editor(s): June 19, 2019
Received by editor(s) in revised form: December 29, 2019
Published electronically: May 8, 2020
Additional Notes: This work was supported by a grant (approval #11871429) from the National Natural Science Foundation of China.
The first author is the corresponding author.
Article copyright: © Copyright 2020 American Mathematical Society