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1. Manipulating elastic waves via engineered spatial curvature profiles

   https://doi.org/10.1117/12.2612573  

   Mazzotti, Matteo ; Gupta, Mohit ; Santangelo, Christian ; Ruzzene, Massimo ( April 2022 , SPIE Smart Structures + Nondestructive Evaluation)

   Fromme, Paul ; Su, Zhongqing (Ed.)

   We investigate curved surfaces operating as geodesic lenses for elastic waves. Consistently with findings in optics, we show that wave propagation occurs along rays that correspond to the geodesics of the curved surfaces, and we establish the geometric equivalence between Gaussian curvature and refractive index. This equivalence is formulated for flexural waves in curved shells by showing that, in the short wavelength limit, the ray equation corresponds to the classical equation of geodesics. We leverage this result to identify a non-Euclidean transformation that maps the geometric profile of a isotropic curved waveguide into a spatially varying refractive index distribution for a planar waveguide. These theoretical predictions are validated first through numerical simulations, and subsequently through experiments on 3D printed curved membranes with different curvature distributions. Numerical and experimental findings confirm that focal regions and caustic networks are correctly predicted based on geodesic evaluations. Our results form the basis for the design of curved profiles that correspond to spatial distributions of the refractive index and induce focal points by forcing waves to propagate along predefined trajectories. The findings of this study also suggest curvature as an attractive alternative to strategies based on the local tailoring of material properties and geometrical patterns that have gained in popularity for gradient-index lens design. 

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2. A relaxed approach for curve matching with elastic metrics

   https://doi.org/10.1051/cocv/2018053  

   Bauer, Martin ; Bruveris, Martins ; Charon, Nicolas ; Møller-Andersen, Jakob ( January 2019 , ESAIM: Control, Optimisation and Calculus of Variations)

   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 family of H 2 -metrics with constant coefficients and scale-invariant H 2 -metrics on both open and closed immersed curves. These families include as particular cases the class of first-order elastic metrics. An essential difference with prior approaches is the way that boundary constraints are dealt with. By leveraging varifold-based similarity metrics we propose a relaxed variational formulation for the matching problem that avoids the necessity of optimizing over the reparametrization group. Furthermore, we show that we can also quotient out finite-dimensional similarity groups such as translation, rotation and scaling groups. The different properties and advantages are illustrated through numerical examples in which we also provide a comparison with related diffeomorphic methods used in shape registration. 

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3. Perturbative expansions of the conformation tensor in viscoelastic flows

   https://doi.org/10.1017/jfm.2018.777  

   Hameduddin, Ismail ; Gayme, Dennice F. ; Zaki, Tamer A. ( January 2019 , Journal of Fluid Mechanics)

   We consider the problem of formulating perturbative expansions of the conformation tensor, which is a positive definite tensor representing polymer deformation in viscoelastic flows. The classical approach does not explicitly take into account that the perturbed tensor must remain positive definite – a fact that has important physical implications, e.g. extensions and compressions are represented similarly to within a negative sign, when physically the former are unbounded and the latter are bounded from below. Mathematically, the classical approach assumes that the underlying geometry is Euclidean, and this assumption is not satisfied by the manifold of positive definite tensors. We provide an alternative formulation that retains the conveniences of classical perturbation methods used for generating linear and weakly nonlinear expansions, but also provides a clear physical interpretation and a mathematical basis for analysis. The approach is based on treating a perturbation as a sequence of successively smaller deformations of the polymer. Each deformation is modelled explicitly using geodesics on the manifold of positive definite tensors. Using geodesics, and associated geodesic distances, is the natural way to model perturbations to positive definite tensors because it is consistent with the manifold geometry. Approximations of the geodesics can then be used to reduce the total deformation to a series expansion in the small perturbation limit. We illustrate our approach using direct numerical simulations of the nonlinear evolution of Tollmien–Schlichting waves. 

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4. Geometric mechanics of ordered and disordered kirigami

   https://doi.org/10.1098/rspa.2022.0822  

   Chaudhary, G. ; Niu, L. ; Han, Q. ; Lewicka, M. ; Mahadevan, L. ( June 2023 , Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   The presence of incomplete cuts in a thin planar sheet can dramatically alter its mechanical and geometrical response to loading, as the cuts allow the sheet to deform strongly in the third dimension, most beautifully demonstrated in kirigami art-forms. We use numerical experiments to characterize the geometric mechanics of kirigamized sheets as a function of the number, size and orientation of cuts. We show that the geometry of mechanically loaded sheets can be approximated as a composition of simple developable units: flats, cylinders, cones and compressed Elasticae. This geometric construction yields scaling laws for the mechanical response of the sheet in both the weak and strongly deformed limit. In the ultimately stretched limit, this further leads to a theorem on the nature and form of geodesics in an arbitrary kirigami pattern, consistent with observations and simulations. Finally, we show that by varying the shape and size of the geodesic in a kirigamized sheet, we can control the deployment trajectory of the sheet, and thence its functional properties as an exemplar of a tunable structure that can serve as a robotic gripper, a soft light window or the basis for a physically unclonable device. Overall our study of disordered kirigami sets the stage for controlling the shape and shielding the stresses in thin sheets using cuts. 

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5. The $$L^p$$-Fisher–Rao metric and Amari–C̆encov $$\alpha $$-Connections

   https://doi.org/10.1007/s00526-024-02660-5  

   Bauer, Martin ; Le Brigant, Alice ; Lu, Yuxiu ; Maor, Cy ( February 2024 , Calculus of Variations and Partial Differential Equations)

   We introduce a family of Finsler metrics, called the$$L^p$$$L^p$-Fisher–Rao metrics$$F_p$$$F_p$, for$$p\in (1,\infty )$$$p\in (1,\infty )$, which generalizes the classical Fisher–Rao metric$$F_2$$$F_2$, both on the space of densities$${\text {Dens}}_+(M)$$${\text{Dens}}_{+}\left(M\right)$and probability densities$${\text {Prob}}(M)$$$\text{Prob}\left(M\right)$. We then study their relations to the Amari–C̆encov$$\alpha $$$\alpha$-connections$$\nabla ^{(\alpha )}$$${\nabla }^{\left(\alpha \right)}$from information geometry: on$${\text {Dens}}_+(M)$$${\text{Dens}}_{+}\left(M\right)$, the geodesic equations of$$F_p$$$F_p$and$$\nabla ^{(\alpha )}$$${\nabla }^{\left(\alpha \right)}$coincide, for$$p = 2/(1-\alpha )$$$p=2/(1-\alpha )$. Both are pullbacks of canonical constructions on$$L^p(M)$$$L^p\left(M\right)$, in which geodesics are simply straight lines. In particular, this gives a new variational interpretation of$$\alpha $$$\alpha$-geodesics as being energy minimizing curves. On$${\text {Prob}}(M)$$$\text{Prob}\left(M\right)$, the$$F_p$$$F_p$and$$\nabla ^{(\alpha )}$$${\nabla }^{\left(\alpha \right)}$geodesics can still be thought as pullbacks of natural operations on the unit sphere in$$L^p(M)$$$L^p\left(M\right)$, but in this case they no longer coincide unless$$p=2$$$p=2$. Using this transformation, we solve the geodesic equation of the$$\alpha $$$\alpha$-connection by showing that the geodesic are pullbacks of projections of straight lines onto the unit sphere, and they always cease to exists after finite time when they leave the positive part of the sphere. This unveils the geometric structure of solutions to the generalized Proudman–Johnson equations, and generalizes them to higher dimensions. In addition, we calculate the associate tensors of$$F_p$$$F_p$, and study their relation to$$\nabla ^{(\alpha )}$$${\nabla }^{\left(\alpha \right)}$.

    

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