The energy minimization problem associated to uniform, isotropic, linearly elastic rods leads to a geometric variational problem for the rod centerline, whose solutions include closed, knotted curves. We give a complete description of the space of closed and quasiperiodic solutions. The quasiperiodic curves are paramet…
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48 results for “elastic rods”
We derive a dimensionally-reduced limit theory for an -dimensional nonlinear elastic body that is slender along dimensions. The starting point is to view an elastic body as an -dimensional Riemannian manifold together with a not necessarily isometric -immersion in -dimensional Euclidean space. The…
New energy model reveals knotted rod configurations.
problem Modeling elastic rods with stretch or inflation.
method Introduced a generalized functional on framed curves, used correspondence to infinite-dimensional Grassmann manifold.
result Explicit parameterizations of all periodic critical framed curves.
Study p-Willmore disks with boundary energies, finding equilibrium configurations.
problem Finding equilibrium configurations for p-Willmore disks with boundary energies.
method Model boundary as Kirchhoff elastic rod, interior term dependent on mean and Gaussian curvatures. Study among topological disks and p-Willmore examples.
result Equilibrium configurations for p-Willmore disks with boundary energies.
Bicycle paths form geodesics in 3D subspaces, related to Kirchhoff rods.
problem Optimizing bicycle paths between two points.
method Variational equations and geometric analysis of bicycle paths.
result Bicycle geodesics are contained in 3D subspaces and relate to Kirchhoff rods.
A new method represents rod shapes as paths in special Euclidean algebra.
problem Representing the shapes of rods and framed curves for mechanical analysis.
method Representing shapes as paths in the special Euclidean algebra.
result The method avoids expensive reconstruction and interpolation in rod mechanics.
The study connects curvature to the elastic energy of non-Euclidean thin bodies.
problem Understanding the elastic energy scaling of non-Euclidean thin bodies.
method Calculating the Γ-limit for the elastic energies of small balls, proving the scaling is \(h^4\).
result The natural scaling for non-Euclidean rods is \(h^4\), confirming previous claims.
The elastic energy functional of a thin elastic rod or sheet is generalized to the case of an M-dimensional manifold in N-dimensional space. We derive potentials for the stress field and curvatures and find the generalized von Karman equations for a manifold in elastic equilibrium. We perform a scaling analysis of an M…
On sphere-filling ropesmath.GT
What is the longest rope on the unit sphere? Intuition tells us that the answer to this packing problem depends on the rope's thickness. For a countably infinite number of prescribed thickness values we construct and classify all solution curves. The simplest ones are similar to the seamlines of a tennis ball, others e…
The Backlund transformation for pseudospherical surfaces, which is equivalent to that of the sine-Gordon equation, can be restricted to give a transformation on space curves that preserves constant torsion. We study its effects on closed curves (in particular, elastic rods) that generate multiphase solutions for the vo…
New geometric mechanics approach to elastic curves.
problem Understanding elastic curves in mechanics and geometry.
method Developed a new geometric mechanics perspective on elastic curves.
result Elastic curves are critical points of length under fixed area and volume constraints.
Upper and lower bounds for hyperbolic rod complements in 3-torus volumes.
problem Understanding geometric properties of hyperbolic rod complements in 3-torus.
method Provided upper and lower bounds for volumes in terms of rod parameters.
result Volume bounds for hyperbolic rod complements in 3-torus depend on rod parameters.
Complete classification of rod complements in 3-torus using topology.
problem Classifying rod complements in the 3-torus.
method Topological arguments.
result Complete classification of all rod complements in the 3-torus.
We combine and extend the work of Alexander & Antman \cite{alexander.82} and Fuller \cite{fuller.71,fuller.78} to give a framework within which precise definitions can be given of topological and geometrical quantities characterising the contortion of open rods undergoing large deformations under end loading. We use th…
Rod flow models Adam's behavior at the edge of stability.
problem Modeling adaptive gradient methods like Adam at the edge of stability.
method Extended rod flow to Adam, considering parameters, first moment, and second moment as variables.
result Rod flow accurately tracks Adam's behavior through the edge-of-stability regime.
Study of rod packings in 3-torus using 3-manifold geometry.
problem Understanding crystal structures in crystallography through rod packings in 3-torus.
method Use of 3-manifold geometry and topology to analyze complements of rod packings.
result Find families of complements that are hyperbolic and Seifert fibred.
Study toric gravitational instantons using rod structures and inequalities.
problem Classify toric ALE/ALF instantons.
method Express signature in terms of rod structure, apply Hitchin-Thorpe inequalities, analyze rod structures with three turning points.
result Necessary conditions for rod structures of toric ALE/ALF instantons.
4-dim intrinsic (material) Riemannian metric of the material 4-D space-time continuum is utilized as the characteristic of the aging processes developing in the material. Manifested through variation of basic material characteristics such as density, moduli of elasticity, yield stress, strength, and toughness.,…
There are many industrial situations where rods are used to stir a fluid, or where rods repeatedly stretch a material such as bread dough or taffy. The goal in these applications is to stretch either material lines (in a fluid) or the material itself (for dough or taffy) as rapidly as possible. The growth rate of mater…
Study evaluates profitability of Islamic banks in Bangladesh using ROA, ROE, and ROD.
problem Evaluating profitability of Islamic banks in Bangladesh.
method Used ROA, ROE, and ROD as measures, analyzed relationships with AU and OE.
result ROD significantly associated with ROA, but not with OE and AU.
The paper tackles learning energies from time-evolving critical points.
problem Learnability of energies from data of critical points.
method Formulates a variational problem and uses Gamma-convergence arguments.
result Minimal solutions from finite observations converge to the exact energy.
A new ODE model explains gradient descent dynamics near edge of stability.
problem Understanding gradient-based training over non-convex landscapes.
method Rod Flow, a new ODE approximation of GD dynamics.
result Rod Flow accurately predicts critical sharpness threshold and self-stabilization in quartic potentials.
ROD reconstructs conditional mean surfaces from observational data, invariant to term order.
problem Regression estimates from observational data can be biased by multicollinearity and term order.
method Retrospective Orthogonal Design (ROD) reconstructs surfaces on a probability-balanced lattice, preserving observed and completing unsupported cells.
result ROD outperformed polynomial regression across various data-generating processes, achieving high out-of-sample .
New method stabilizes tensegrity structures suitable for engineering.
problem Tensegrity structures often have unstable modes unsuitable for engineering.
method Proposes a relationship between rods and strings for full-rank convexity.
result Designs a stable three-rod three-string tensegrity.
Skew parallelogram nets factorize, encompassing discrete differential geometry.
problem Factorization of polynomials in discrete differential geometry.
method Lax representation, Bäcklund transformations, factorization of polynomials.
result Skew parallelogram nets encompass all systems with polynomial representations.
Proves uniqueness and existence of toric gravitational instantons.
problem Proves uniqueness and existence of four-dimensional asymptotically flat, Ricci-flat, toric gravitational instantons.
method Adapting black hole uniqueness theorems to a harmonic map formulation of Ricci-flat metrics with torus symmetry.
result Proves that instantons are uniquely characterised by their rod structure and that for every admissible rod structure, there exists a smooth instanton.
Weaved helices form mechanically stable 3D structures.
problem Creating stable 3D structures from helical elements.
method Exploiting screw symmetry and invariant cylindrical rod packing to form triply periodic arrangements.
result Demonstrated nineteen triply periodic arrangements of interwoven helices.
Closed-form relations and approximations for SE(3) derivatives for robust numerical simulations.
problem Deriving closed-form derivatives and approximations for SE(3) for robust numerical simulations.
method Avoiding block partitioning, deriving higher-order approximations for differential, first and second derivatives, Jacobian, and Hessian.
result Compact and numerically robust closed-form relations for SE(3) derivatives.
The correspondence of stationary, axisymmetric, asymptotically flat space-times and bundles over a reduced twistor space has been established in four dimensions. The main impediment for an application of this correspondence to examples in higher dimensions is the lack of a higher-dimensional equivalent of the Ernst pot…
Paper introduces untangling number to quantify 3-periodic tangle complexity.
problem Quantifying the complexity of 3-periodic tangles in biological, chemical, and physical systems.
method Introduces untangling number, a measure of minimum distance to ground state through diagrammatic operations.
result For infinite open curves, generic ground states are crystallographic rod packings.
This paper develops a new method for constructing splines on Lie groups using Poisson equation solutions.
problem Existing methods for constructing splines on Lie groups have limitations and assumptions that may not reflect actual curves.
method The paper introduces a new approach using solutions of the Poisson equation on Lie groups to construct splines.
result The new method allows for global splines with arbitrary initial conditions, improving curve reconstruction.
This review discusses solutions to Einstein's equations using twistor theory.
problem Finding solutions to Einstein's vacuum equations using twistor theory.
method Holomorphic vector bundles on twistor space and patching matrices.
result Holomorphic patching matrix is simpler than the metric and determines the rod structure.
Sparse elasticity reconstruction from local displacements reduces error.
problem Reconstructing elasticity from limited data.
method Sparse elasticity reconstruction theory, local clustering, alternating optimization.
result Higher spatial resolution elasticity distribution estimation.
The paper generalizes the second Pappus-Guldin theorem for calculating volumes of bodies.
problem Calculating the volume of a body cut into perpendicular slices.
method Using a generalized formula and properties of centroids and floating bodies.
result A curve with centroid property exists for convex bodies, leading to simpler volume calculations.
Analyzes properties of stiffness tensors for elastic wave imaging.
problem Characterizing stiffness tensor fields for elastic wave imaging.
method Finsler-geometric methods applied to anisotropic stiffness tensor fields.
result Conditions for Finsler-geometric methods to be applicable.
Elastic Cashq-fin.GN
Elastic Cash adjusts money supply to stabilize interest rates.
problem Stabilizing interest rates in a decentralized system.
method Modifies supply to keep interest rate fixed by public market.
result Improves elasticity of US Dollar and new cryptocurrencies.
We find a Weierstrass-like formula for 2D elastic maps.
problem Understanding elastic maps between planar domains.
method Develop a Weierstrass representation for critical points of certain energy functionals.
result Elastic maps admit a Weierstrass representation in terms of holomorphic functions.
Proposes using elastic demand to improve forecasting accuracy.
problem Mismatch between planned supply and actual demand due to demand variance.
method Reallocate historical elastic demand to reduce forecasting variance.
result Improves forecasting and supply planning effectiveness.
Characterizes null Lagrangians in Cosserat elasticity.
problem Understanding null Lagrangians in micropolar elasticity.
method Applying Olver and Sivaloganathan's theorem to characterize null Lagrangians.
result Complete characterization of null Lagrangians for three-dimensional bodies and shells.
Approximate 3D elastic curves with exact constraints
problem Designing and approximating 3D elastic curves
method Numerically stable method for recovering 11 parameters
result Fast and stable approximation of arbitrary curves
New algorithm selects genes for cancer classification using adaptive elastic net and conditional mutual information.
problem Selecting informative genes for microarray cancer classification.
method Adaptive Elastic Net with Conditional Mutual Information (AEN-CMI).
result AEN-CMI achieves the best classification performance with fewer genes.
Stable discretizations for elastic flow on Riemannian manifolds.
problem Discretizing elastic flow on curved spaces.
method Conformally flat Riemannian manifolds discretization.
result Robust and quadratic convergence of the method.
The paper studies rigidity and continuity in nonlinear elasticity on manifolds and hypersurfaces.
problem Rigidity and continuity properties of elastic bodies in non-Euclidean settings.
method Geometric rigidity estimates, asymptotic rigidity of elastic membranes, simplified geometric proof of continuous dependence.
result Established geometric rigidity estimate and proved asymptotic rigidity of elastic membranes.
Study preserves planar and graphical properties of curves under elastic flow.
problem Maintaining planar and graphical properties of non-compact curves under elastic flow.
method Extended recent work on adapted elastic energy to derive thresholds for planar and graphical embeddedness.
result Derived new Li--Yau type inequality for complete planar curves.
The paper studies the free elastic flow of closed curves and finds their asymptotic shape converges to a circle.
problem Challenges in studying the asymptotic behavior of the free elastic flow for closed curves.
method Analysis of the free elastic flow as an -gradient flow for Euler's elastic energy.
result An appropriate rescaling of initial curves geometrically close to circles converges to a unique round circle.
Elastic stripsmath.DG
Motivated by the problem of finding an explicit description of a developable narrow Moebius strip of minimal bending energy, which was first formulated by M. Sadowsky in 1930, we will develop the theory of elastic strips. Recently E.L. Starostin and G.H.M. van der Heijden found a numerical description for an elastic Mo…
Migrating elastic flowsmath.AP
Study on migrating elastic flows of curves across half-planes.
problem Migrating elastic flows of curves from upper to lower half-planes.
method Analytical and numerical construction of migrating elastic flows.
result Construction of various migrating elastic flows.
Study gauge freedoms in elastic wave equations and Dirichlet-to-Neumann map.
problem Recover stiffness tensor and density from Dirichlet-to-Neumann map.
method Analyze invariance under coordinate transformations and gauge freedoms.
result Present gauge freedoms in the Dirichlet-to-Neumann map for Riemannian elastic wave equation.