58 research outputs found
Topology counts: force distributions in circular spring networks
Filamentous polymer networks govern the mechanical properties of many
biological materials. Force distributions within these networks are typically
highly inhomogeneous and, although the importance of force distributions for
structural properties is well recognized, they are far from being understood
quantitatively. Using a combination of probabilistic and graph-theoretical
techniques we derive force distributions in a model system consisting of
ensembles of random linear spring networks on a circle. We show that
characteristic quantities, such as mean and variance of the force supported by
individual springs, can be derived explicitly in terms of only two parameters:
(i) average connectivity and (ii) number of nodes. Our analysis shows that a
classical mean-field approach fails to capture these characteristic quantities
correctly. In contrast, we demonstrate that network topology is a crucial
determinant of force distributions in an elastic spring network.Comment: 5 pages, 4 figures. Missing labels in Fig. 4 added. Reference fixe
Geodesics in Heat
We introduce the heat method for computing the shortest geodesic distance to
a specified subset (e.g., point or curve) of a given domain. The heat method is
robust, efficient, and simple to implement since it is based on solving a pair
of standard linear elliptic problems. The method represents a significant
breakthrough in the practical computation of distance on a wide variety of
geometric domains, since the resulting linear systems can be prefactored once
and subsequently solved in near-linear time. In practice, distance can be
updated via the heat method an order of magnitude faster than with
state-of-the-art methods while maintaining a comparable level of accuracy. We
provide numerical evidence that the method converges to the exact geodesic
distance in the limit of refinement; we also explore smoothed approximations of
distance suitable for applications where more regularity is required
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