Menghan Wang

Optimal decomposition and recombination of isostatic geometric constraint systems for designing layered materials

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 this http URL)

Algorithm 951: Cayley Analysis of Mechanism Configuration Spaces using CayMos: Software Functionalities and Architecture

For a common class of two-dimensional (2D) mechanisms called 1-dof tree-decomposable linkages, we present a software package, CayMos, which uses new theoretical results from Sitharam and Wang [2014] and Sitharam et al. [2011a, 2011b] to implement efficient algorithmic solutions for (a) meaningfully representing and visualizing the connected components in the Euclidean realization space; (b) finding a path of continuous motion between two realizations in the same connected component, with or without restricting the realization type (sometimes called orientation type); and (c) finding two “closest” realizations in different connected components.

Symmetry in Sphere-Based Assembly Configuration Spaces

Many remarkably robust, rapid and spontaneous self-assembly phenomena in nature can be modeled geometrically starting from a collection of rigid bunches of spheres. This paper highlights the role of symmetry in sphere-based assembly processes. Since spheres within bunches could be identical and bunches could be identical as well, the underlying symmetry groups could be of large order that grows with the number of participating spheres and bunches. Thus, understanding symmetries and associated isomorphism classes of microstates correspond to various types of macrostates can significantly reduce the complexity of computing entropy and free energy, as well as paths and kinetics, in high dimensional configuration spaces. In addition, a precise understanding of symmetries is crucial for giving provable guarantees of algorithmic accuracy and efficiency in such computations. In particular, this may aid in predicting crucial assembly-driving interactions. This is a primarily expository paper that develops a novel, original framework for dealing with symmetries in configuration spaces of assembling spheres with the following goals. (1) We give new, formal definitions of various concepts relevant to sphere-based assembly that occur in previous work, and in turn, formal definitions of their relevant symmetry groups leading to the main theorem concerning their symmetries. These previously developed concepts include, for example, (a) assembly configuration spaces, (b) stratification of assembly configuration space into regions defined by active constraint graphs, (c) paths through the configurational regions, and (d) coarse assembly pathways. (2) We demonstrate the new symmetry concepts to compute sizes and numbers of orbits in two example settings appearing in previous work. (3) We give formal statements of a variety of open problems and challenges using the new conceptual definitions.

Combinatorial Rigidity and Independence ofźGeneralized Pinned Subspace-Incidence Constraint Systems

Given a hypergraph H with m hyperedges and a set X of mpins, i.e. globally fixed subspaces in Euclidean space

Technical note: How the Beast really moves: Cayley analysis of mechanism realization spaces using CayMos

\mathbb {R}^{d}

Cayley configuration spaces of 2D mechanisms, Part I: extreme points, continuous motion paths and minimal representations

Rd, a pinned subspace-incidence system is the pair H,i¾źX, with the constraint that each pin in X lies on the subspace spanned by the point realizations in

Cayley Configuration Spaces of 1-dof Tree-decomposable Linkages, Part I: Structure and Extreme Points

\mathbb {R}^d