Discrete and nonlocal models of Engesser and Haringx elastica

International audienceIn this paper, a generalized discrete elastica including both bending and shear elastic interactions is developed and its possible link with nonlocal beam continua is revealed. This lattice system can be viewed as the generalization of the Hencky bar-chain model, which can be retrieved in the case of infinite shear stiffness. The shear contribution in the discrete elastica is introduced by following the approach of Engesser (normal and shear forces are aligned with and perpendicular to the link axis, respectively) and that of Haringx (shear force is parallel to end section of links), both supported by physical arguments. The nonlinear analysis of the shearable-bendable discrete elastica under axial load is accomplished. Buckling and post-buckling of the lattice systems are analyzed in a geometrically exact framework. The buckling loads of both the discrete Engesser and Haringx elastica are analytically calculated, and the post-buckling behavior is numerically studied for large displacement. Nonlocal Timoshenko-type beam models, including both bending and shear stiffness, are then built from the continualization of the discrete systems. Analytical solutions for the fundamental buckling loads of the nonlocal Engesser and Haringx elastica models are given, and their first post-budding paths are numerically computed and compared to those of the discrete Engesser and Haringx elastica. It is shown that the nonlocal Timoshenko-type beam models efficiently capture the scale effects associated with the shearable-bendable discrete elastica

Fairing of Discrete Planar Curves by Discrete Euler's Elasticae

After characterizing the integrable discrete analogue of the Euler's elastica, we focus our attention on the problem of approximating a given discrete planar curve by an appropriate discrete Euler's elastica. We carry out the fairing process via a $L^2\!$-distance minimization to avoid the numerical instabilities. The optimization problem is solved via a gradient-driven optimization method (IPOPT). This problem is non-convex and the result strongly depends on the initial guess, so that we use a discrete analogue of the algorithm provided by Brander et al., which gives an initial guess to the optimization method

Higher-Order Gradient Elasticity Models Applied to Geometrically Nonlinear Discrete Systems

The buckling and post-buckling behavior of a nonlinear discrete repetitive system, the discrete elastica, is studied herein. The nonlinearity essentially comes 'from the geometrical effect, whereas the constitutive law of each component is reduced to linear elasticity. The paper primarily :focuses on the relevancy of higher -order continuum approximations of the difference equations, also called continualization of the lattice model. The pseudo-differential operator of the lattice equations are expanded by Taylor series, up to the second or the ifourth-order, leading to an equivalent second-order or ifourth-order gradient elasticity model. The accuracy of each of these models is compared to the initial lattice model and to some other approximation methods based on a rational expansion of the pseudo-differential operator. It is found, as anticipated, that the higher level of truncation is chosen, the better accuracy is obtained with respect to the lattice solution. This paper also outlines the key role played by the boundary conditions, which also need to be consistently continualized from their discrete expressions. It is concluded I hat higher-order gradient elasticity Triode's can efficiently capture the scale etre( is of lattice models

Efficient Regularization of Squared Curvature

Curvature has received increased attention as an important alternative to length based regularization in computer vision. In contrast to length, it preserves elongated structures and fine details. Existing approaches are either inefficient, or have low angular resolution and yield results with strong block artifacts. We derive a new model for computing squared curvature based on integral geometry. The model counts responses of straight line triple cliques. The corresponding energy decomposes into submodular and supermodular pairwise potentials. We show that this energy can be efficiently minimized even for high angular resolutions using the trust region framework. Our results confirm that we obtain accurate and visually pleasing solutions without strong artifacts at reasonable run times.Comment: 8 pages, 12 figures, to appear at IEEE conference on Computer Vision and Pattern Recognition (CVPR), June 201

B\'ezier curves that are close to elastica

We study the problem of identifying those cubic B\'ezier curves that are close in the L2 norm to planar elastic curves. The problem arises in design situations where the manufacturing process produces elastic curves; these are difficult to work with in a digital environment. We seek a sub-class of special B\'ezier curves as a proxy. We identify an easily computable quantity, which we call the lambda-residual, that accurately predicts a small L2 distance. We then identify geometric criteria on the control polygon that guarantee that a B\'ezier curve has lambda-residual below 0.4, which effectively implies that the curve is within 1 percent of its arc-length to an elastic curve in the L2 norm. Finally we give two projection algorithms that take an input B\'ezier curve and adjust its length and shape, whilst keeping the end-points and end-tangent angles fixed, until it is close to an elastic curve.Comment: 13 pages, 15 figure