arXiv research

Unified treatment of elastic metrics for curves in any dimension.

Study uniquely determines Riemannian metric derivatives from boundary data.

New metrics for surface shapes incorporating curve properties.

In this paper, the elastic Dirichlet-to-Neumann map $Ξ_g$ is studied for the stationary elasticity system in a compact Riemannian manifold $(Ω,g)$ with smooth boundary $\partial Ω$. By overcoming methodological difficulties, we explicitly get matrix-valued full symbol for the elastic Dirichlet-to-Neumann map $Ξ_g$. We …

In this article we introduce a family of elastic metrics on the space of parametrized surfaces in 3D space using a corresponding family of metrics on the space of vector valued one-forms. We provide a numerical framework for the computation of geodesics with respect to these metrics. The family of metrics is invariant …

In this article, the clarification to Note 4 (arXiv:1202.0941) for n=8 is considered. In this connection, answers to the following questions are given. 1. How to classify the metric hypercomplex orthogonal group alternative-elastic algebras for n=8? 2. How to associate the metric hypercomplex orthogonal group alternati…

Study geodesics in constrained curve spaces, including elastic curves and concentric circles.

The chapter reviews metrics for comparing curves, focusing on quotient elastic and square root velocity metrics.

Non-Euclidean, or incompatible elasticity is an elastic theory for pre-stressed materials, which is based on a modeling of the elastic body as a Riemannian manifold. In this paper we derive a dimensionally-reduced model of the so-called membrane limit of a thin incompatible body. By generalizing classical dimension red…

Classical elasticity is concerned with bodies that can be modeled as smooth manifolds endowed with a reference metric that represents local equilibrium distances between neighboring material elements. The elastic energy associated with a configuration of a body in classical elasticity is the sum of local contributions …

Metric anomalies arising from a distribution of point defects (intrinsic interstitials, vacancies, point stacking faults), thermal deformation, biological growth, etc. are well known sources of material inhomogeneity and internal stress. By emphasizing the geometric nature of such anomalies we seek their representation…

Study of elastic models in non-Euclidean spaces via Γ-convergence.

The paper proves rigidity for shells in non-Euclidean spaces.

A new numerical framework simplifies elastic surface matching and comparison.

In the shape analysis approach to computer vision problems, one treats shapes as points in an infinite-dimensional Riemannian manifold, thereby facilitating algorithms for statistical calculations such as geodesic distance between shapes and averaging of a collection of shapes. The performance of these algorithms depen…

Theory models nonlinear soft tissue elasticity and remodeling using extended Finsler geometry.

We study the pull-back of the 2-parameter family of quotient elastic metrics introduced in Mio-Srivastava-Joshi on the space of arc-length parameterized loops. This point of view has the advantage of concentrating on the manifold of arc-length parameterized curves, which is a very natural manifold when the analysis of …

This paper introduces a new mathematical formulation and numerical approach for the computation of distances and geodesics between immersed planar curves. Our approach combines the general simplifying transform for first-order elastic metrics that was recently introduced by Kurtek and Needham, together with a relaxatio…

Improved performance in shape identification tasks using elastic metrics in t-SNE and UMAP.

We derive a dimensionally-reduced limit theory for an $n$-dimensional nonlinear elastic body that is slender along $k$ dimensions. The starting point is to view an elastic body as an $n$-dimensional Riemannian manifold together with a not necessarily isometric $W^{1,2}$-immersion in $n$-dimensional Euclidean space. The…

Study explores kinematics of surfaces under metric restrictions.

We derive a continuum model for incompatible elasticity as a variational limit of a family of discrete nearest-neighbor elastic models. The discrete models are based on discretizations of a smooth Riemannian manifold $(M,\mathfrak{g})$, endowed with a flat, symmetric connection $\nabla$. The metric $\mathfrak{g}$ deter…

Geometrically reformulates elasticity theory using exterior calculus.

In this paper we study a class of Riemannian metrics on the space of unparametrized curves and develop a method to compute geodesics with given boundary conditions. It extends previous works on this topic in several important ways. The model and resulting matching algorithm integrate within one common setting both the …

We prove a relation between the scaling $h^β$ of the elastic energies of shrinking non-Euclidean bodies $S_h$ of thickness $h\to 0$, and the curvature along their mid-surface $S$. This extends and generalizes similar results for plates [BLS16, LRR] to any dimension and co-dimension. In particular, it proves that the na…

Analyzes properties of stiffness tensors for elastic wave imaging.

New method for surface analysis using restricted deformation bases.

This paper introduces an elasticity reconstruction method based on local displacement observations of elastic bodies. Sparse reconstruction theory is applied to formulate the underdetermined inverse problems of elasticity reconstruction including unobserved areas. An online local clustering scheme called a superelement…

The edge of torn elastic sheets and growing leaves often form a hierarchical buckling pattern. Within non-Euclidean plate theory this complex morphology can be understood as low bending energy isometric immersions of hyperbolic Riemannian metrics. With this motivation we study the isometric immersion problem in strip a…

This paper describes a novel framework for computing geodesic paths in shape spaces of spherical surfaces under an elastic Riemannian metric. The novelty lies in defining this Riemannian metric directly on the quotient (shape) space, rather than inheriting it from pre-shape space, and using it to formulate a path energ…

Elastic Cash adjusts money supply to stabilize interest rates.

4-dim intrinsic (material) Riemannian metric $G$ of the material 4-D space-time continuum $P$ 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.,…

Characterizes null Lagrangians in Cosserat elasticity.

Approximate 3D elastic curves with exact constraints

The paper studies rigidity and continuity in nonlinear elasticity on manifolds and hypersurfaces.

Study preserves planar and graphical properties of curves under elastic flow.

The paper studies the free elastic flow of closed curves and finds their asymptotic shape converges to a circle.

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…

Study on migrating elastic flows of curves across half-planes.

Study gauge freedoms in elastic wave equations and Dirichlet-to-Neumann map.

Solves curve migration problem with elastic flows.

New insights into stability of special curves on spheres.

Due to the advantage of achieving a better performance under weak regularization, elastic net has attracted wide attention in statistics, machine learning, bioinformatics, and other fields. In particular, a variation of the elastic net, adaptive elastic net (AEN), integrates the adaptive grouping effect. In this paper,…

Symmetric elastic knots are found for certain classes with dihedral symmetry.

We study a class of elastic energy functionals for maps between planar domains (among them the so-called squared distance functional) whose critical points (elastic maps) allow a far more complete theory than one would expect from general elasticity theory. For some of these functionals elastic maps even admit a "Weier…

New discrete curves defined in space forms with geometric properties.

Demand variance can result in a mismatch between planned supply and actual demand. Demand shaping strategies such as pricing can be used to shift elastic demand to reduce the imbalance. In this work, we propose to consider elastic demand in the forecasting phase. We present a method to reallocate the historical elastic…

The elastic flow, which is the $L^2$-gradient flow of the elastic energy, has several applications in geometry and elasticity theory. We present stable discretizations for the elastic flow in two-dimensional Riemannian manifolds that are conformally flat, i.e.\ conformally equivalent to the Euclidean space. Examples in…