Algorithm 990

Abstract

For configurations of point-sets that are pairwise constrained by distance intervals, the EASAL software implements a suite of algorithms that characterize the structure and geometric properties of the configuration space. The algorithms generate, describe, and explore these configuration spaces using generic rigidity properties, classical results for stratification of semi-algebraic sets, and new results for efficient sampling by convex parametrization. The article reviews the key theoretical underpinnings, major algorithms, and their implementation. The article outlines the main applications such as the computation of free energy and kinetics of assembly of supramolecular structures or of clusters in colloidal and soft materials. In addition, the article surveys select experimental results and comparisons.

Realizations of Isostatic Material Frameworks

“…Such characterizations exist even for frameworks whose bar lengths are in non-Euclidean, polyhedral norms, and are strongly linked to the concept of flattenability [49,50] of graphs, characterized by forbidden minor subgraphs, for example, partial 2-trees are exactly those graphs that forbid the complete subgraphs on four vertices, or K 4 . Convex Cayley parameterization has been used for analyzing sphere-based assembly configuration spaces in previous studies, [51][52][53] further applied to predicting crucial interactions in virus assembly in previous studies. [54,55] The idea is to drop sufficiently many edges from a glassy framework (arbitrarily large versions of the Trihex), shown in red in Figure 9, so that the remaining graph has a convex Cayley configuration space (is a partial 2-tree) parameterized by nonedge lengths shown in green.…”

“…Such characterizations exist even for frameworks whose bar lengths are in non‐Euclidean, polyhedral norms, and are strongly linked to the concept of flattenability [ 49,50 ] of graphs, characterized by forbidden minor subgraphs, for example, partial 2‐trees are exactly those graphs that forbid the complete subgraphs on four vertices, or . Convex Cayley parameterization has been used for analyzing sphere‐based assembly configuration spaces in previous studies, [ 51–53 ] further applied to predicting crucial interactions in virus assembly in previous studies. [ 54,55 ]…”

Rigidity for sticky discs

“…The authors of [34] conjecture that Question 5.6 has a positive answer. Interestingly, it conceivable that the Maxwell counting heuristic is sufficient for generic rigidity for generic radius ball packing.…”

“…In a series of papers, Connelly and co-workers [9,10,15] relate configurations that are locally maximally dense to the rigidity of a related tensegrity (see, e.g., [39]) over the contact graph. Notably, the recent work in [13] proves results about the number of contacts appearing in such locally maximally dense packings under appropriate genericity assumptions.Another motivation comes from geometric constraint solving [34], where combinatorial methods are also applied to structures made of disks and spheres.Laman's Theorem Given the relationship between disk packings and associated frameworks, it is very tempting to go further and apply the methods of combinatorial rigidity (see, e.g., [25]) theory to infer geometric or physical properties from the contact graph alone. Several recent works in the soft matter literature [16,21,30] use such an approach.The combinatorial approach is attractive because we have a very good understanding of framework rigidity in dimension 2, provided that p is not very degenerate.…”

“…Another motivation comes from geometric constraint solving [34], where combinatorial methods are also applied to structures made of disks and spheres.…”