Study on surfaces minimizing elastic energy with boundary constraints.
problem Finding stable configurations of surfaces with elastic boundaries and surface energy.
method Investigation of critical surfaces with mean curvature and spontaneous curvature, coupled to boundary elastic energy.
result Characterization and minimization of surface energy for specific topological shapes.
Study of red blood cells using elastic surface theory.
problem Understanding the shape of red blood cells.
method Used Helfrich-Canham functional to model red blood cells as elastic surfaces.
result Cassinian ovals, except for the round sphere, do not solve the shape equation.
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
The paper studies surfaces with spherical curvature lines and their generation by constrained elastic curves.
problem Understanding surfaces with spherical curvature lines and their generation mechanisms.
method The approach involves Lie sphere transformations, Legendre curves, and polynomial conserved quantities of connections.
result Lie applicable surfaces with exactly one family of spherical curvature lines are generated by the lift of constrained elastic curves.
In this paper, we consider the classical variational problem in the Galilean space. we develop the Euler-Lagrange equations for a elastic line on an oriented surface in the Galilean 3-dimensional space G3. Using the varia- tion method, we will try to give some characterization for the solution curve (the elastic lin…
New method weaves paper strips for designing curved surfaces with elasticity.
problem Designing general curved surfaces with geometrical elasticity.
method Shape optimization of paper strips using nonlinear elasticity theory.
result Demonstrated creation of catenoid and helicoid surfaces with 54 paper strips.
New metrics for surface shapes incorporating curve properties.
problem Developing metrics for surface shape spaces.
method Incorporates geodesic and normal curvatures of curves on surfaces.
result Explicitly defined 6-parameter family of metrics.
A new method for 3D surface registration using dynamic programming.
problem Elastic shape registration of 3D surfaces.
method Optimization over a subset of reparametrizations using dynamic programming.
result Proposes an algorithm that produces a solution closer to optimal than gradient-based methods.
Paper proves existence of solutions for complex surface diffusion equation.
problem Existence of solutions for anisotropic surface diffusion with elasticity.
method Cahn-Taylor minimizing movement scheme for three-dimensional analysis.
result Proves existence of classical solutions without curvature regularization.
Study on surface configurations with curvature and elasticity.
problem Equilibrium configurations of surfaces with curvature and elasticity.
method Investigates the Euler-Helfrich functional, focusing on axially symmetric surfaces and their variational problems.
result Critical surfaces for the Euler-Helfrich functional, if axially symmetric, satisfy a simpler second order variational problem.
A new numerical framework simplifies elastic surface matching and comparison.
problem Challenging problem in surface comparison and matching in computer vision.
method Relaxing the geodesic boundary constraint using a varifold fidelity metric.
result Flexibility to deal with arbitrary topologies and sampling patterns, scalability to large meshes.
We develop a theory of axisymmetric surfaces minimizing a combination of surface tension and nematic elastic energies which may be suitable for describing simple film and bubble shapes. As a function of the elastic constant and the applied tension on the bubbles, we find the analogues of the unduloid, sphere, and nodoi…
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…
Study on the Euler-Plateau energy with elastic modulus, focusing on minimizers and critical surfaces.
problem Minimizing the Euler-Plateau energy with elastic modulus.
method Analyzing the energy functional and its minimizers, considering different boundary conditions and topological constraints.
result Potential minimizers are highly dependent on physical rigidity parameters, and the area of critical surfaces can be computed from boundary data.
We introduce elastic geodesic grids for easy-to-fabricate, deployable structures.
problem Approximating freeform surfaces with deployable structures.
method Geodesic curves on target surfaces, kinematic mechanism, differential geometry.
result Elastic geodesic grids can approximate freeform surfaces easily and deployably.
Computes elastic grids that approximate 3D surfaces without physical simulations.
problem Creating planar grids that fit complex 3D surfaces efficiently.
method Uses differential geometry to minimize bending energy and nestle to the surface.
result Elastic grids can approximate 3D surfaces without physical simulations.
Researchers reconstruct stiffness tensors from limited data in anisotropic elasticity.
problem Reconstructing stiffness tensors from partial data around one polarization.
method Using algebraic geometry and slowness surfaces, the approach leverages the algebraic geometry of families of slowness surfaces.
result For tensors in a dense open subset, a small amount of data around one polarization uniquely determines the entire slowness surface and stiffness tensor.
Study shows global invertibility in nonlinear elasticity with vanishing self-repulsion term.
problem Global invertibility in nonlinear elasticity with a vanishing nonlocal self-repulsion term.
method Proves global invertibility in the Γ-limit of elastic energy with a vanishing nonlocal self-repulsion term. result Global invertibility can be obtained in the Γ-limit of the elastic energy with a vanishing nonlocal self-repulsion term. Let α(s) be an arc on a connected oriented surface S in E3, parameterized by arc length s, with torsion τ and length l. The total square torsion F of α is defined by T=\int_{0}^{l}τ^{2}ds\ $. . The arc α is called a relaxed elastic line of second kind if it is an extremal for the variational problem of minimizing the v…
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 …
Classifies soap film surfaces with vertical potentials.
problem Classifying soap film surfaces with vertical potentials.
method Variational characterization of n-elastic curves. result Obtains a full description of n-elastic curves. Study on bending energy of surfaces with curvature concentration, deriving new lower bounds.
problem Analyzing the Willmore energy of surfaces with curvature concentration.
method Using isoperimetric inequalities and framed loops, derive new lower bounds for the bending energy.
result Optimal blowup rates of the Willmore energy when curvature is concentrated.
Study geodesics in constrained curve spaces, including elastic curves and concentric circles.
problem Geodesics in constrained curve spaces with Sobolev metrics.
method Intrinsic and constructive approaches.
result Construct geodesics in elastic curve and concentric circle spaces.
Two classification theorems for Willmore surfaces in S² × S².
problem Classifying Willmore surfaces in S² × S².
method Analytical proofs for minimal and product type surfaces.
result Classification of Willmore surfaces in S² × S².
Paper proves rigidity estimates for hyperbolic shells and applies them to \(Γ\)-limit theory.
problem Rigidity of hyperbolic shells and their \(Γ\)-limit behavior.
method Nonlinear rigidity estimates for \(H^1\) deformations and hyperbolic shells with clamped lateral boundary.
result Derives the optimal exponent \(h^{-4/3}\) for hyperbolic shells.
Let α be an arc on a connected oriented surface S in Minkowski 3-space, parameterized by arc length s, with torsion τ and length l. The total square torsion H of α is defined by . The arc is called a relaxed elastic line of second kind if it is an extremal for the variational prob…
Characterizes neutral deformation modes of minimal surfaces.
problem Understanding the energy content of deformation modes of minimal surfaces.
method Analyzes the energy content of stretching, drilling, and bending modes of minimal surfaces.
result All isometries of a minimal surface are globally neutral and give rise to soft elasticity.
Study of p-biharmonic curves and their properties.
problem Generalizing biharmonic curves to p-biharmonic curves. method Classification and analysis of p-biharmonic curves on surfaces and space forms. result Existence and stability of p-biharmonic curves on closed surfaces. Study explores kinematics of surfaces under metric restrictions.
problem Understanding the kinematics of surfaces under metric constraints.
method Analyzed three energy contents: stretching, drilling, and bending.
result Metric restrictions can hinder the elastic response of a shell.
Helical ribbons arise in many biological and engineered systems, often driven by anisotropic surface stress, residual strain, and geometric or elastic mismatch between layers of a laminated composite. A full mathematical analysis is developed to analytically predict the equilibrium deformed helical shape of an initiall…
For a bounded domain Ω⊂Rn with smooth boundary, we explicitly calculate the first two coefficients of the asymptotic expansion of the trace of the strongly continuous semigroup associated with the Navier-Lamé operator on Ω as t→0+. These coefficients (i.e., spectral invariants) provide precise …
New method for surface analysis using restricted deformation bases.
problem Surface registration and comparison without pre-registered data.
method Elastic Riemannian metrics with basis-restricted transformations.
result Effective implementation on human body and face scans.
This paper puts forth a new formulation and algorithm for the elastic matching problem on unparametrized curves and surfaces. Our approach combines the frameworks of square root normal fields and varifold fidelity metrics into a novel framework, which has several potential advantages over previous works. First, our var…
Li-Yau inequality applied to curves in 2D space.
problem Curves in 2D space with low elastic energy.
method Classical Li-Yau inequality applied to curves.
result Analogous results for curves in 2D space with low elastic energy.
Novel defects in hyperbolic sheets explain complex wrinkling patterns in nature.
problem Understanding complex wrinkling patterns in thin elastic hyperbolic surfaces.
method Non-Euclidean plate theory and investigation of branch points.
result Branch points are natural defects in hyperbolic sheets, influencing their morphology robustly.
For a given family of smooth closed curves γ1,...,γα⊂R3 we consider the problem of finding an elastic \emph{connected} compact surface M with boundary γ=γ1∪...∪γα. This is realized by minimizing the Willmore energy W on a suitable class of competitors. While the direct minimi…
The wave equation utt=c2uxx is generally regarded as a linear approximation to the equation describing the amplitude of a transversely vibrating elastic string in the plane. But, as is shown in \cite{BC96}, the assumption of transverse vibration in fact implies that the wave equation describes the vibration…
Sophie Germain's mean curvature deserves recognition as a surface shape measure.
problem Identifying the shape of a surface using curvature measurements.
method Characterizing surface shape through principal curvatures and their averages.
result Mean curvature should be named after Sophie Germain.
This work presents a general unified theory for coupled nonlinear elastic and inelastic deformations of curved thin shells. The coupling is based on a multiplicative decomposition of the surface deformation gradient. The kinematics of this decomposition is examined in detail. In particular, the dependency of various ki…
We prove a relation between the scaling hβ of the elastic energies of shrinking non-Euclidean bodies Sh of thickness h→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…
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…
The paper proves rigidity for shells in non-Euclidean spaces.
problem Proving rigidity for shells in non-Euclidean spaces.
method Analyzing a stretching plus bending functional of an elastic shell in a Riemannian manifold.
result A sequence of immersions of asymptotically vanishing energy converges to an isometric immersion of the shell.
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.
The Poisson problem consists in finding an immersed surface Σ⊂Rm minimising Germain's elastic energy (known as Willmore energy in geometry) with prescribed boundary, boundary Gauss map and area which constitutes a non-linear model for the equilibrium state of thin, clamped elastic plates originating f…
New method shortens and straightens curves, proving convergence and well-posedness.
problem Shortening and straightening of curves.
method Conceptual shift in curve shortening to tangent aligning, variational study of geometric flows.
result Proves convergence to a straight line and global well-posedness for various geometric flows.
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…
Transforms curves and surfaces for efficient geometric analysis.
problem Efficiently analyzing and comparing curves and surfaces.
method Square root velocity transformation for curves and intrinsic comparison for surfaces.
result Fundamental geometric properties of curves under the transformation.
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.