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Paolo Piersanti

Publications and source records attributed to Paolo Piersanti.

14 recordsLinked to original sources
arXiv2026

On the feasible set of a membrane shell under confinement. Smooth inactive admissible displacements are dense

We establish a density property for the obstacle problem of a linearly elastic elliptic membrane shell required to remain confined in a prescribed half-space. For a bounded Lipschitz reference domain, and under the sole assumption that the reference configuration lies at a strictly positive distance from the confining plane, we prove that smooth admissible displacements for which the confinement constraint is inactive (satisfied with a strictly positive margin) are dense in the set of all admissible displacements. This removes the additional condition on the unit normal to the mid-surface required in previous work, and the approximation scheme is elementary and adaptable to other unilateral obstacle problems.

math.AP↗
arXiv2026

Existence of solutions for time-dependent Signorini-type problems in linearised viscoelasticity

In this paper we establish the existence of solutions for a model describing the evolution of a linearly viscoelastic body which is constrained to remain confined in a prescribed half-space. The confinement condition under consideration is of Signorini type, and is given over the boundary of the linearly viscoelastic body under consideration. We show that one such variational problem admits solutions and we coin a novel concept of solution which, differently from the available literature, is valid even in the case where the viscoelastic body starts its motion in contact with the obstacle. Additionally, under additional assumptions on the constituting material, we show that when the applied body force is lifted the deformed linearly viscoelastic body returns to its rest position at an exponential rate of decay.

math.AP↗
arXiv2026

Diffuse Adiabatic Flows in Thermally Coupled Grounded Shallow Ice Sheets: Modelling and Analysis

In this article we propose a novel thermodynamical model which couples the evolution of the thickness of a grounded shallow ice sheet with the evolution of its internal temperature. Both the grounded shallow ice sheet surface elevation and the ice internal temperature are subjected to physical constraints. The equations governing the evolution of the grounded shallow ice sheet thickness are degenerate, and the ice internal temperature evolves in a moving domain. First, we formally model the phenomenon under consideration by adopting strategies akin to those employed in the construction of diffuse-interface models. Second, we establish the existence of solutions for one such formal model by means of the penalty method, and we observe that the low regularity of the problem under consideration prevents us from obtaining a standard concept of solution.

math.AP↗
arXiv2026

On the justification of Koiter's model for generalised membrane shells of the "first kind'' confined in a half-space

In this paper we justify Koiter's model for linearly elastic generalised membrane shells of the first kind subjected to remaining confined in a prescribed half-space. After showing that the confinement condition considered in this paper is in general stronger, under the validity of the Kirchhoff-Love assumptions, than the classical Signorini condition, we formulate the corresponding obstacle problem for a three-dimensional linearly elastic generalised membrane shell of the first kind, and we conduct a rigorous asymptotic analysis as the thickness approaches to zero on the unique solution for one such model. We show that the solution to the three-dimensional obstacle problem converges to the unique solution of a two-dimensional model consisting of a set of variational inequalities that are posed over the abstract completion of a non-empty, closed and convex set. We recall that the two-dimensional limit model obtained departing from Koiter's model is characterised by the same variational inequalities as the two-dimensional model obtained departing from the variational formulation of a three-dimensional linearly elastic generalised membrane shell subjected to remaining confined in a prescribed half-space, with the remarkable difference that the sets where solutions for these two-dimensional limit models are sought do not coincide, in general. In order to complete the justification of Koiter's model for linearly elastic generalised membrane shells of the first kind subjected to remaining confined in a half-space, we give sufficient conditions ensuring that the sets where solutions for these two-dimensional limit models are sought coincide.

math.AP↗
arXiv2025

Geometrical mixed finite element methods for fourth order obstacle problems in linearised elasticity

This paper is devoted to the study of a novel mixed Finite Element Method for approximating the solutions of fourth order variational problems subjected to a constraint. The first problem we consider consists in establishing the convergence of the error of the numerical approximation of the solution of a biharmonic obstacle problem. The contents of this section are meant to generalise the approach originally proposed by Ciarlet \& Raviart, and then complemented by Ciarlet \& Glowinski. The second problem we consider amounts to studying a two-dimensional variational problem for linearly elastic shallow shells subjected to remaining confined in a prescribed half-space. We first study the case where the parametrisation of the middle surface for the linearly elastic shallow shell under consideration has non-zero curvature, and we observe that the numerical approximation of this model via a mixed Finite Element Method based on conforming elements requires the implementation of the additional constraint according to which the gradient matrix of the dual variable has to be symmetric. However, differently from the biharmonic obstacle problem previously studied, we show that the numerical implementation of this result cannot be implemented by solely resorting to Courant triangles. Finally, we show that if the middle surface of the linearly elastic shallow shell under consideration is flat, the symmetry constraint required for formulating the constrained mixed variational problem announced in the second part of the paper is not required, and the solution can thus be approximated by solely resorting to Courant triangles. The theoretical results we derived are complemented by a series of numerical experiments.

math.NA↗
arXiv2025

On the justification of Koiter's model for elliptic membranes subjected to an interior normal compliance contact condition

The purpose of this paper is twofold. First, we rigorously justify Koiter's model for linearly elastic elliptic membrane shells in the case where the shell is subject to a geometrical constraint modelled via a normal compliance contact condition defined in the interior of the shell. To achieve this, we establish a novel density result for non-empty, closed, and convex subsets of Lebesgue spaces, which are applicable to cases not covered by the ``density property'' established in [Ciarlet, Mardare \& Piersanti, \emph{Math. Mech. Solids}, 2019]. Second, we demonstrate that the solution to the two-dimensional obstacle problem for linearly elastic elliptic membrane shells, subjected to the interior normal compliance contact condition, exhibits higher regularity throughout its entire definition domain. A key feature of this result is that, while the transverse component of the solution is, in general, only of class $L^2$ and its trace is \emph{a priori} undefined, the methodology proposed here, partially based on [Ciarlet \& Sanchez-Palencia, \emph{J. Math. Pures Appl.}, 1996], enables us to rigorously establish the well-posedness of the trace for the transverse component of the solution by means of an \emph{ad hoc} formula.

math.AP↗
arXiv2025

Nonconvex optimization and convergence of stochastic gradient descent, and solution of asynchronous game

We review convergence and behavior of stochastic gradient descent for convex and nonconvex optimization, establishing various conditions for convergence to zero of the variance of the gradient of the objective function, and presenting a number of simple examples demonstrating the approximate evolution of the probability density under iteration, including applications to both classical two-player and asynchronous multiplayer games

math.OC↗
arXiv2024

On the dynamics of shallow ice sheets. Modelling and analysis

We formulate a model describing the evolution of thickness in a shallow ice sheet lying over a lithosphere. The model is thus governed by a set of variational inequalities that involve nonlinearities in the time derivative and in the elliptic term. Solutions are constructed via a finite difference scheme in time.

math.AP↗
arXiv2024

Numerical approximation of the solution of Koiter's model for an elliptic membrane shell subjected to an obstacle via the penalty method

This paper is devoted to the analysis of a numerical scheme based on the Finite Element Method for approximating the solution of Koiter's model for a linearly elastic elliptic membrane shell subjected to remaining confined in a prescribed half-space. First, we show that the solution of the obstacle problem under consideration is uniquely determined and satisfies a set of variational inequalities which are governed by a fourth order elliptic operator, and which are posed over a non-empty, closed, and convex subset of a suitable space. Second, we show that the solution of the obstacle problem under consideration can be approximated by means of the penalty method. Third, we show that the solution of the corresponding penalised problem is more regular up to the boundary. Fourth, we write down the mixed variational formulation corresponding to the penalised problem under consideration, and we show that the solution of the mixed variational formulation is more regular up to the boundary as well. In view of this result concerning the augmentation of the regularity of the solution of the mixed penalised problem, we are able to approximate the solution of the one such problem by means of a Finite Element scheme. Finally, we present numerical experiments corroborating the validity of the mathematical results we obtained.

math.NA↗
arXiv2023

Numerical approximation of the solution of an obstacle problem modelling the displacement of elliptic membrane shells via the penalty method

In this paper we establish the convergence of a numerical scheme based, on the Finite Element Method, for a time-independent problem modelling the deformation of a linearly elastic elliptic membrane shell subjected to remaining confined in a half space. Instead of approximating the original variational inequalities governing this obstacle problem, we approximate the penalized version of the problem under consideration. A suitable coupling between the penalty parameter and the mesh size will then lead us to establish the convergence of the solution of the discrete penalized problem to the solution of the original variational inequalities. We also establish the convergence of the Brezis-Sibony scheme for the problem under consideration. Thanks to this iterative method, we can approximate the solution of the discrete penalized problem without having to resort to nonlinear optimization tools. Finally, we present numerical simulations validating our new theoretical results.

math.AP↗
arXiv2022

Modelling Virus Contact Mechanics under Atomic Force Imaging Conditions

In this paper we present a discrete model governing the deformation of a convex regular polygon subjected not to cross a given flat rigid surface, on which it initially lies in correspondence of one point only. First, we set up the model in the form of a set of variational inequalities posed over a non-empty, closed and convex subset of a suitable Euclidean space. Secondly, we show the existence and uniqueness of the solution. The model provides a simplified illustration of processes involved in virus imaging by atomic force microscopy: adhesion to a surface, distributed strain, relaxation to a shape that balances adhesion and elastic forces. The analysis of numerical simulations results based on this model opens a new way of estimating the contact area and elastic parameters in virus contact mechanics studies.

math.AP↗
arXiv2022

A three-dimensional discrete model for approximating the deformation of a viral capsid subjected to lying over a flat surface

In this paper we present a three-dimensional discrete model governing the deformation of a viral capsid, modelled as a regular icosahedron and subjected not to cross a given flat rigid surface on which it initially lies in correspondence of one vertex only. First, we set up the model in the form of a set of variational inequalities posed over a non-empty, closed and convex subset of a suitable space. Secondly, we show the existence and uniqueness of the solution for the proposed model. Finally, we numerically test this model and we observe that the outputs of the numerical experiments comply with physics.

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