812 research outputs found
Geometric auxetics
Get PDFWe formulate a mathematical theory of auxetic behavior based on one-parameter
deformations of periodic frameworks. Our approach is purely geometric, relies
on the evolution of the periodicity lattice and works in any dimension. We
demonstrate its usefulness by predicting or recognizing, without experiment,
computer simulations or numerical approximations, the auxetic capabilities of
several well-known structures available in the literature. We propose new
principles of auxetic design and rely on the stronger notion of expansive
behavior to provide an infinite supply of planar auxetic mechanisms and several
new three-dimensional structures
Extremal Configurations of Hinge Structures
Get PDFWe study body-and-hinge and panel-and-hinge chains in R^d, with two marked
points: one on the first body, the other on the last. For a general chain, the
squared distance between the marked points gives a Morse-Bott function on a
torus configuration space. Maximal configurations, when the distance between
the two marked points reaches a global maximum, have particularly simple
geometrical characterizations. The three-dimensional case is relevant for
applications to robotics and molecular structures
Expansive periodic mechanisms
Get PDFA one-parameter deformation of a periodic bar-and-joint framework is
expansive when all distances between joints increase or stay the same. In
dimension two, expansive behavior can be fully explained through our theory of
periodic pseudo-triangulations. However, higher dimensions present new
challenges. In this paper we study a number of periodic frameworks with
expansive capabilities in dimension and register both similarities
and contrasts with the two-dimensional case
Deformations of crystal frameworks
Get PDFWe apply our deformation theory of periodic bar-and-joint frameworks to
tetrahedral crystal structures. The deformation space is investigated in detail
for frameworks modelled on quartz, cristobalite and tridymite
Liftings and stresses for planar periodic frameworks
Get PDFWe formulate and prove a periodic analog of Maxwell's theorem relating
stressed planar frameworks and their liftings to polyhedral surfaces with
spherical topology. We use our lifting theorem to prove deformation and
rigidity-theoretic properties for planar periodic pseudo-triangulations,
generalizing features known for their finite counterparts. These properties are
then applied to questions originating in mathematical crystallography and
materials science, concerning planar periodic auxetic structures and ultrarigid
periodic frameworks.Comment: An extended abstract of this paper has appeared in Proc. 30th annual
Symposium on Computational Geometry (SOCG'14), Kyoto, Japan, June 201
Rotationally-invariant mapping of scalar and orientational metrics of neuronal microstructure with diffusion MRI
Full text linkWe develop a general analytical and numerical framework for estimating intra-
and extra-neurite water fractions and diffusion coefficients, as well as
neurite orientational dispersion, in each imaging voxel. By employing a set of
rotational invariants and their expansion in the powers of diffusion weighting,
we analytically uncover the nontrivial topology of the parameter estimation
landscape, showing that multiple branches of parameters describe the
measurement almost equally well, with only one of them corresponding to the
biophysical reality. A comprehensive acquisition shows that the branch choice
varies across the brain. Our framework reveals hidden degeneracies in MRI
parameter estimation for neuronal tissue, provides microstructural and
orientational maps in the whole brain without constraints or priors, and
connects modern biophysical modeling with clinical MRI.Comment: 25 pages, 12 figures, elsarticle two-colum
Periodic Tilings and Auxetic Deployments
Get PDFWe investigate geometric characteristics of a specific planar periodic framework with three degrees of freedom. While several avatars of this structural design have been considered in materials science under the name of chiral or missing rib models, all previous studies have addressed only local properties and limited deployment scenarios. We describe the global configuration space of the framework and emphasize the geometric underpinnings of auxetic deformations. Analogous structures may be considered in arbitrary dimension
Infinitesimal Periodic Deformations and Quadrics
Get PDFWe describe a correspondence between the infinitesimal deformations of a periodic bar-and-joint framework and periodic arrangements of quadrics. This intrinsic correlation provides useful geometric characteristics. A direct consequence is a method for detecting auxetic deformations, identified by a pattern consisting of homothetic ellipsoids. Examples include frameworks with higher crystallographic symmetry
Auxetics Abounding
Get PDFAuxetic behavior refers to lateral widening upon stretching. Although a structural origin for this rather counterintuitive type of deformation was often suggested, a theoretical understanding of the role of geometry in auxetic behavior has been a challenge for a long time. However, for structures modeled as periodic bar-and-joint frameworks, including atom-and-bond frameworks in crystalline materials, there is a complete geometric solution which opens endless possibilities for new auxetic designs. We construct a large family of three-dimensional auxetic periodic mechanisms and discuss the ideas involved in their design
Singularity Locus for the Endpoint Map of Serial Manipulators with Revolute Joints
Get PDFWe present a theoretical and algorithmic method for describing the singularity locus for the endpoint map of any serial manipulator with revolute joints. As a surface of revolution around the first joint, the singularity locus is determined by its intersection with a fixed plane through the first joint. The resulting plane curve is part of an algebraic curve called the singularity curve. Its degree can be computed from the specialized case of all pairs of consecutive joints coplanar, when the singularity curve is a union of circles, counted with multiplicity two. Knowledge of the degree and a simple iterative procedure for obtaining sample points on the singularity curve lead to the precise equation of the curve. © Springer Science+Business Media Dordrecht 2014
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