arXiv subjects

Lorenzo Cavallina

Publications and source records attributed to Lorenzo Cavallina.

23 records · Page 2Linked to original sources

arXiv2019

The homogenization method for topology optimization of structures: old and new

These are the lecture notes of a short course on the homogenization method for topology optimization of structures, given by Grégoire Allaire, during the "GSIS International Summer School 2018" at Tohoku University (Sendai, Japan). The goal of this course is to review the necessary mathematical tools of homogenization theory and apply them to topology optimization of mechanical structures. The ultimate application, targeted in this course, is the topology optimization of structures built with lattice materials. Practical and numerical exercises are given, based on the finite element free software FreeFem++.

math.AP↗

arXiv2019

Analysis of two-phase shape optimization problems by means of shape derivatives

This is my Ph.D. Thesis at Tohoku Univeristy (July 2018). It presents the theory on shape derivatives and focuses on its applications to two-phase optimization problems. In particular, we treat the two-phase torsion problem and a two-phase analogous of Serrin's overdetermined problem.

math.AP↗

arXiv2018

Stability analysis of the two-phase torsional rigidity near a radial configuration

Let $Ω_0$ denote the unit ball of $\mathbb{R}^N$ ($N\ge 2$) centered at the origin. We suppose that $Ω_0$ contains a core, given by a smaller concentric ball $D_0$, made of a (possibly) different material. We discover that, depending on the relative hardness of the two materials, this radial configuration can either be a local maximizer for the torsional rigidity functional $E$ or a saddle shape. In this paper we consider perturbations that simultaneously act on the boundaries $\partial D_0$ and $\partialΩ_0$. This gives rise to resonance effects that are not present when $\partial D_0$ or $\partialΩ_0$ are perturbed in isolation. A detailed analysis of the sign of the second order shape derivative of $E$ is then made possible by employing the use of spherical harmonics.

math.AP↗

arXiv2017

Locally optimal configurations for the two-phase torsion problem in the ball

We consider the unit ball $Ω\subset \mathbb{R}^N$ ($N\ge2$) filled with two materials with different conductivities. We perform shape derivatives up to the second order to find out precise information about locally optimal configurations with respect to the torsional rigidity functional. In particular we analyse the role played by the configuration obtained by putting a smaller concentric ball inside $Ω$. In this case the stress function admits an explicit form which is radially symmetric: this allows us to compute the sign of the second order shape derivative of the torsional rigidity functional with the aid of spherical harmonics. Depending on the ratio of the conductivities and on the radius of the smaller ball, a symmetry breaking phenomenon occurs.

math.OC↗

arXiv2015

Non-trivial translation-invariant valuations on $L^\infty$

Translation-invariant valuations on the space $L^\infty(\mathbb{R}^n)$ are examined. We prove that such functionals vanish on functions with compact support. Moreover a rich family of non-trivial translation-invariant valuations on $L^\infty(\mathbb{R}^n)$ is constructed through the use of ultrafilters on $\mathbb{R}$.

math.FA↗