influence of the parameterization in the interval solution of elastic beams

We are going to analyze the interval solution of an elastic beam under uncertain boundary conditions. Boundary conditions are defined as rotational springs presenting interval stiffness. Developments occur according to the interval analysis theory, which is affected, at the same time, by the overestimation of interval limits (also known as overbounding, because of the propagation of the uncertainty in the model). We suggest a method which aims to reduce such an overestimation in the uncertain solution. This method consists in a reparameterization of the closed form Euler-Bernoulli solution and set intersection

Current Options for Visualization of Local Deformation in Modern Shape Analysis Applied to Paleobiological Case Studies

In modern shape analysis, deformation is quantified in different ways depending on the algorithms used and on the scale at which it is evaluated. While global affine and non-affine deformation components can be decoupled and computed using a variety of methods, the very local deformation can be considered, infinitesimally, as an affine deformation. The deformation gradient tensor F can be computed locally using a direct calculation by exploiting triangulation or tetrahedralization structures or by locally evaluating the first derivative of an appropriate interpolation function mapping the global deformation from the undeformed to the deformed state. A suitable function is represented by the thin plate spline (TPS) that separates affine from non-affine deformation components. F, also known as Jacobian matrix, encodes both the locally affine deformation and local rotation. This implies that it should be used for visualizing primary strain directions (PSDs) and deformation ellipses and ellipsoids on the target configuration. Using C = FTF allows, instead, one to compute PSD and to visualize them on the source configuration. Moreover, C allows the computation of the strain energy that can be evaluated and mapped locally at any point of a body using an interpolation function. In addition, it is possible, by exploiting the second-order Jacobian, to calculate the amount of the non-affine deformation in the neighborhood of the evaluation point by computing the body bending energy density encoded in the deformation. In this contribution, we present (i) the main computational methods for evaluating local deformation metrics, (ii) a number of different strategies to visualize them on both undeformed and deformed configurations, and (iii) the potential pitfalls in ignoring the actual three-dimensional nature of F when it is evaluated along a surface identified by a triangulation in three dimensions

Local and Global Energies for Shape Analysis in Medical Imaging

In a previous contribution a new Riemannian shape space, named TPS space, was introduced to perform statistics on shape data. This space was endowed with a Rie-mannian metric and a flat connection, with torsion, compatible with the given metric. This connection allows the definition of a Parallel Transport of the deformation compatible with the threefold decomposition in spherical, deviatoric and non affine components. Such a Parallel Transport also conserves the-energy, strictly related to the total elastic strain energy stored by the body in the original deformation. New machinery is here presented in order to calculate the bending energy on the body only (body bending energy) in order to restrict it exclusively within physical boundaries of objects involved in the deformation analysis. The novelty of this new procedure resides in the fact that we propose a new metric to conserve during the TPS direct transport. This allows transporting the shape change more coherently with the mechanical meaning of the deformation. The geometry of the TPS Space is then further developed in order to better represent the relationship between the-energy, the strain energy and the so called bending-energy densities

Seeing the wood through the trees. Combining shape information from different landmark configurations

The geometric morphometric (GM) analysis of complex anatomical structures is an ever more powerful tool to study biological variability, adaptation and evolution. Here, we propose a new method (combinland), developed in R, meant to combine the morphological information contained in different landmark coordinate sets into a single dataset, under a GM context. combinland builds a common ordination space taking into account the entire shape information encoded in the starting configurations. We applied combinland to a Primate case study including 133 skulls belonging to 14 species. On each specimen, we simulated photo acquisitions converting the 3D landmark sets into six 2D configurations along standard anatomical views. The application of combinland shows statistically negligible differences in the ordination space compared to that of the original 3D objects, in contrast to a previous method meant to address the same issue. Hence, we argue combinland allows to correctly retrieve 3D-quality statistical information from 2D landmark configurations. This makes combinland a viable alternative when the extraction of 3D models is not possible, recommended, or too expensive, and to make full use of disparate sources (and views) of morphological information regarding the same specimens. The code and examples for the application of combinland are available in the Arothron R package

The TPS Direct Transport: a new method for transporting deformations in the Size-and-shape Space

Modern shape analysis allows the fine comparison of shape changes occurring between different objects. Very often the classic machineries of Generalized Procrustes Analysis and Principal Component Analysis are used in order to contrast the shape change occurring among configurations represented by homologous landmarks. However, if size and shape data are structured in different groups thus constituting different morphological trajectories, a data centering is needed if one wants to compare solely the deformation representing the trajectories. To do that, inter-individual variation must be filtered out. This maneuver is rarely applied in studies using simulated or real data. A geometrical procedure named Parallel Transport, that can be based on various connection types, is necessary to perform such kind of data centering. Usually, the Levi Civita connection is used for interpolation of curves in a Riemannian space. It can also be used to transport a deformation. We demonstrate that this procedure does not preserve some important characters of the deformation, even in the affine case. We propose a novel procedure called `TPS Direct Transport' which is able to perfectly transport deformation in the affine case and to better approximate non affine deformation in comparison to existing tools. We recommend to center shape data using the methods described here when the differences in deformation rather than in shape are under study

The decomposition of deformation: new metrics to enhance shape analysis in medical imaging

In landmarks-based Shape Analysis size is measured, in most cases, with Centroid Size. Changes in shape are decomposed in affine and non affine components. Furthermore the non affine component can be in turn decomposed in a series of local deformations (partial warps). If the extent of deformation between two shapes is small, the difference between centroid size and m-Volume increment is barely appreciable. In medical imaging applied to soft tissues bodies can undergo very large deformations, involving large changes in size. The cardiac example, analyzed in the present paper, shows changes in m-Volume that can reach the 60%. We show here that standard Geometric Morphometrics tools (landmarks, Thin Plate Spline, and related decomposition of the deformation) can be generalized to better describe the very large deformations of biological tissues, without losing a synthetic description. In particular, the classical decomposition of the space tangent to the shape space in affine and non affine components is enriched to include also the change in size, in order to give a complete description of the tangent space to the size-and-shape space. The proposed generalization is formulated by means of a new Riemannian metric describing the change in size as change in m-Volume rather than change in Centroid Size. This leads to a redefinition of some aspects of the Kendall’s size-and-shape space without losing Kendall’s original formulation. This new formulation is discussed by means of simulated examples using 2D and 3D platonic shapes as well as a real example from clinical 3D echocardiographic data. We demonstrate that our decomposition based approaches discriminate very effectively healthy subjects from patients affected by Hypertrophic Cardiomyopathy

Capturing the helical to spiral transitions in thin ribbons of nematic elastomers

We provide a quantitative description of the helicoid-to-spiral transition in thin ribbons of nematic elastomers using an elementary calculation based on a Koiter-type plate with incompatible reference configuration. Our calculation confirms that such transition is ruled by the competition between stretching energy and bending energy

Capturing the helical to spiral transitions in thin ribbons of nematic elastomers

We provide a quantitative description of the helicoid-to-spiral transition in thin ribbons of nematic elastomers using an elementary calculation based on a Koiter-type plate with incompatible reference configuration. Our calculation confirms that such transition is ruled by the competition between stretching energy and bending energy