We consider the minimization of integral functionals in one dimension and their approximation by r-adaptive finite elements. Including the grid of the FEM approximation as a variable in the minimization, we are able to show that the optimal grid configurations have a well-defined limit when the number of nodes in the grid is being sent to infinity. This is done by showing that the suitably...
We propose a two-scale model to resolve essential features of developmental tissue deformations. The model couples individual cellular behavior to the mechanics at tissue scale. This is realized by a multiphase-field model addressing the motility, deformability and interaction of cells on an evolving surface. The surface evolution is due to bending elasticity, with bending properties...
A number of strategies have been proposed to overcome the constraint imposed by Gauss Egregium theorem regarding morphing of a thin sheet. These strategies often boil down to devising a metasurface whose subunits can accommodate non-uniform in-plane stretch, e.g. pneumatic channels, swellable materials, kirigami or origami patterns. I will discuss a different approach inspired in the way...
"The current wave of artificial intelligence is transforming industry, society, and the sciences at an unprecedented pace. Yet, despite its remarkable progress, today’s AI still suffers from two major limitations: a lack of reliability and excessive energy consumption.
This lecture will begin with an overview of this dynamic field, focusing first on reliability. We will present recent...
Inflatable morphing matter represents a frontier in programmable architecture and soft robotics, enabling dramatic shape changes driven by simple pressure inputs. In this talk, I will present a unified vision for how instabilities and geometric design can be harnessed to create inflatable systems that morph, lock, and reconfigure on demand. Starting from the fundamentals of buckling, snapping,...
I will discuss some of our recent works on structure-preserving deep learning for the design of neural networks with specific properties - such as non-expansiveness or 1-Lipschitz regularity - and their application to imaging and to the solution of partial differential equations.
Arnol'd pointed out that Euler's equations for an inviscid incompressible fluid can be seen as geodesic equations on the manifold of volume-preserving maps, when endowed with the ambient $L^2$ metric. On the opposite, overdamped, end of fluid dynamics, a density driven two-phase flow in a porous medium can be interpreted as a gradient flow of the potential energy with respect to a metric that...
The publication of Disquisitiones circa superficies curvas (1828) is widely regarded as marking the beginning of modern differential geometry. Although important results in the geometry of curves and surfaces had already been achieved during the 18th century, Gauss’s contribution inaugurated an entirely new phase in the development of the discipline.
The composition of the Disquisitiones...
We study the morphing of 3D objects within the framework of non-linear elasticity with large distortions. A distortion field induces a target metric, and the configuration which is effectively realized by a material body is the one that minimizes the distance, measured through the elastic energy, between the target metric and the actual one.
Morphing through distortions might have a...
The Cahn-Hilliard energy is a celebrated phase-field model for the smooth approximation of the area of domain’s boundaries. Its L2 gradient flow provides an excellent approximation, both theoretically and numerically, of the smooth mean curvature flow.
In this talk, I will present a new model for approximating the area of general interfaces not associated with any interior domain, which we...
Ribbons are elastic bodies that are thin and narrow. Many ribbons in nature, from seed pods to molecular assemblies, have a non-trivial internal geometry, making them incompatible with Euclidean space. In many cases, this results in shape transitions between narrow and wide ribbons with the same internal geometry. In this talk we will show how this phenomenon can be explained mathematically in...
A classical result (‘Liouville’s theorem’) states that a sufficiently regular map u in Euclidean space whose differential Du belongs to the group SO(n) of orientation preserving isometries at every point is affine. The quantitative version of the result states that for maps of a bounded connected set U with Lipschitz boundary the L2 distance of the differential Du from a constant can be...
