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Studying the changes of shape is a common concern in many scientific fields. We address here two problems: (1) quantifying the deformation between two given shapes and (2) transporting this deformation to morph a third shape. These operations can be done with or without point correspondence, depending on the availability of a surface matching algorithm, and on the type of mathematical procedure adopted. In computer vision, the re-targeting of emotions mapped on faces is a common application. We contrast here four different methods used for transporting the deformation toward a target once it was estimated upon the matching of two shapes. These methods come from very different fields such as computational anatomy, computer vision and biology. We used the large diffeomorphic deformation metric mapping and thin plate spline, in order to estimate deformations in a deformational trajectory of a human face experiencing different emotions. Then we use naive transport (NT), linear shift (LS), direct transport (DT) and fanning scheme (FS) to transport the estimated deformations toward four alien faces constituted by 240 homologous points and identifying a triangulation structure of 416 triangles. We used both local and global criteria for evaluating the performance of the 4 methods, e.g., the maintenance of the original deformation. We found DT, LS and FS very effective in recovering the original deformation while NT fails under several aspects in transporting the shape change. As the best method may differ depending on the application, we recommend carefully testing different methods in order to choose the best one for any specific application.

In the problem of the synthesis of metamaterials, the pantographic architecture revealed remarkable potentialities. Indeed it allowed for the synthesis of second gradient 2D (nonlinear) continua: i.e. 2D shells whose deformation energy depends also on the second derivatives of displacements in the tangent directions to the reference configuration. Moreover, pantographic architecture seems to be able to produce metamaterials whose macroscopic elongations are large, albeit remaining in the elastic regime. The theoretically shown potentialities have started to become of ≫practical≪ interest thanks to a series of experiments, which were made possible by the recent 3D additive manufacturing. The actual construction of pantographic architecture has been based on the design of two arrays of beams interconnected by small cylinders, whose behavior can be modeled in different ways: if they are very short they can be regarded as clamps, while if they are short enough as elastic (or inelastic for large rotations) cylindrical hinges connecting the beams of different arrays. Otherwise, they must be modeled as elastic (or inelastic) elements allowing for relative rotations and displacements. In this paper, we focus on this particular case and we introduce, after a homogenization based on heuristic arguments, a 2D generalized continuum model whose kinematics is characterized by two placement and rotation fields (one for each array of beams) and whose deformation energy depends on relative displacements and rotations. The offset between the two beams arrays is proven to be an essential tool for defining effective invariant kinematical deformation measures. In facts, one wants to postulate a deformation energy for the introduced 2D generalized continuum which gives predictions in agreement with those given by the more refined 3D model where the pantographic architecture is described with its maximum geometric complexity and where the constituting material is assumed to be modelable as a standard 3D first gradient continuum. In the present paper, in order to arrive at the correct conjecture for the postulated energy, we consider the concept of averages of rotations in SO(3) Lie group. The used enriched kinematics is seen to be a possible alternative to the adoption of second gradient 2D models. Some rather surprising deformation processes are studied, where interesting non-symmetric post-buckling phenomena are observed in both the models used. Mentioned post-buckling has been observed experimentally.

Transporting deformations from a template to a different one is a typical task of the shape analysis. In particular, it is necessary to perform such a kind of transport when performing group-wise statistical analyses in Shape or Size and Shape Spaces. A typical example is when one is interested in separating the difference in function from the difference in shape. The key point is: given two different templates (Formula presented.) and (Formula presented.) both undergoing their own deformation, and describing these two deformations with the diffeomorphisms (Formula presented.) and (Formula presented.), then when is it possible to say that they are experiencing the same deformation? Given a correspondence between the points of (Formula presented.) and (Formula presented.) (i.e. a bijective map), then a naïve possible answer could be that the displacement vector (Formula presented.), associated to each corresponding point couple, is the same. In this manuscript, we assume a different viewpoint: two templates undergo the same deformation if for each corresponding point couple of the two templates the condition (Formula presented.) holds or, in other words, the local metric (non linear strain) induced by the two diffeomorphisms is the same for all the corresponding points.

A nonlinear staged construction analysis is adopted here to simulate the forming process of timber gridshell. In this respect, particular attention is paid to the modelling of the cylindrical joints connecting the orthogonal laths. Cylindrical joints enable the initially flat lattice to be formed into a doubly curved gridshell shape; therefore, accurate modelling of the joints’ mechanical behaviour is a crucial aspect when simulating the forming process of timber gridshells. Another important parameter influencing the gridshell shape is the orientation of laths making up the initially flat lattice. Using an equivalent continuum definition, the final geometry of the gridshells, obtained from different laths orientations, is analysed in here to assess how laths orientation affects the structural shape in terms of funicularity. The Relaxed Funicularity Ellipse method presented in Gabriele et al. (2018) is used for such purpose.

We exploit the possibility of deforming a solid volume, or a surface as well, by assigning a target metric. As well known, an elastic solid may change its shape under two different kinds of actions: one are the loadings, the other one are the distortions, aka, the pre-strains. Actually, a distortion prescribes the target metric sought by the solid, and the metric effectively realized is the one that minimize the distance, measured through an elastic energy, between the target metric and the actual one. The challenge of our work is to use the metric information taken from a pair of configurations X ¯ , X of a given body B x , to deform a different body B y , so that its deformation be as similar as possible to the one suffered by B x . In particular, we assess the metric change between the two configurations X ¯ , X of the template B x ; then, we use this information to build a target metric for B y , exploiting the notion of distortions: given a source configuration Y ¯ , we find a target configuration Y by minimizing an appropriate elastic energy. The proposed method is very effective in deforming solids as well as surfaces, and in reproducing similar deformations even when applied to different bodies.

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 Riemannian 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 three-fold decomposition in spherical, deviatoric, and non-affine components. Such a parallel transport also conserves the Γ-energy, strictly related to the total elastic strain energy stored by the body in the original deformation. A new approach is here presented in order to calculate the bending energy on the body alone (body bending energy) and 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 be preserved 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 discussed in order to better represent the relationship between the Γ-energy, the strain energy, and the so-called bending energy densities.

We present and discuss a method to infer noninvasively information on the fibre architecture in real human left ventricle walls. The method post-processes the echocardiographic data acquired by three-dimensional speckle tracking echocardiography through a MatLab-based protocol, already presented, discussed and validated. Our results indicate the difference between the role of endocardial and epicardial principal strain lines, at the systolic peak, and set the bases for possible future investigations aimed to analyse the onset of specific cardiac diseases through noninvasive analysis of LV fibre architecture. Moreover, we set a rational statistical analysis to investigate the reliability of the proposed method.

A thin rectangular plate is modelled as an (initially flat) shell. Following Koiter, the two fundamental forms of the deformed middle surface are then used to define the strain measures of the body. On the middle surface of the plate two local coordinates are introduced: we will call them longitudinal and transversal, respectively. It is assumed that the components of the displacement field which characterize the middle surface kinematics can be expressed as a product of two functions: one defined along the longitudinal coordinate and one defined along the transversal coordinate. Given an explicit expression of the latter functions, the 2D plate fields are reduced to 1D ones. The functions of the transversal coordinate are chosen so that the stretch along it together with the membrane shear vanish. It is worth noting that the linearization of these constraints leads to the well-known Vlasov’s assumptions. It is shown that by modelling each side of a thin walled beam as a 1D continuum, the entire assembly can be reduced to a 1D model as well. This procedure gives rise to an hyperelastic 1D beam model in which at least the warping effect is taken into account. The main features of the model are shown by means of some simple applications.