Optimal Decomposition and Recombination of Isostatic Geometric Constraint Systems for Designing Layered Materials

TL;DR

This paper introduces an efficient algorithm for optimal recursive decomposition of certain geometric constraint systems, enabling better design and analysis of layered materials like silica bilayers and microfibrils.

Contribution

It presents an O(n^3) algorithm for a broad class of isostatic or underconstrained systems using a new canonical DR-plan concept, improving decomposition and realization processes.

Findings

Algorithm achieves optimal decomposition in polynomial time.

Application to modeling layered materials like silica bilayers.

Software implementation and demonstrations are publicly available.

Abstract

Optimal recursive decomposition (or DR-planning) is crucial for analyzing, designing, solving or finding realizations of geometric constraint sytems. While the optimal DR-planning problem is NP-hard even for general 2D bar-joint constraint systems, we describe an O(n^3) algorithm for a broad class of constraint systems that are isostatic or underconstrained. The algorithm achieves optimality by using the new notion of a canonical DR-plan that also meets various desirable, previously studied criteria. In addition, we leverage recent results on Cayley configuration spaces to show that the indecomposable systems---that are solved at the nodes of the optimal DR-plan by recombining solutions to child systems---can be minimally modified to become decomposable and have a small DR-plan, leading to efficient realization algorithms. We show formal connections to well-known problems such as completion of underconstrained systems. Well suited to these methods are classes of constraint systems that can be used to efficiently model, design and analyze quasi-uniform (aperiodic) and self-similar, layered material structures. We formally illustrate by modeling silica bilayers as body-hyperpin systems and cross-linking microfibrils as pinned line-incidence systems. A software implementation of our algorithms and videos demonstrating the software are publicly available online (visit http://cise.ufl.edu/~tbaker/drp/index.html.)