Analytic dependence on parameters for Evans' approximated Weak KAM solutions

We consider a variational principle for approximated Weak KAM solutions proposed by Evans. For Hamiltonians in quasi-integrable form, we prove that the map which takes the parameters to Evans\u2019 approximated solution is real analytic. In the mechanical case, we compute a recursive system of periodic partial differential equations identifying univocally the coefficients for the power series of the perturbative parameter

The Dirichlet problem in a planar domain with two moderately close holes

We investigate a Dirichlet problem for the Laplace equation in a domain of $\mathbb{R}^2$ with two small close holes. The domain is obtained by making in a bounded open set two perforations at distance $|\epsilon_1|$ one from the other and each one of size $|\epsilon_1\epsilon_2|$. In such a domain, we introduce a Dirichlet problem and we denote by $u_{\epsilon_1,\epsilon_2}$ its solution. We show that the dependence of $u_{\epsilon_1,\epsilon_2}$ upon $(\epsilon_1,\epsilon_2)$ can be described in terms of real analytic maps of the pair $(\epsilon_1,\epsilon_2)$ defined in an open neighborhood of $(0,0)$ and of logarithmic functions of $\epsilon_1$ and $\epsilon_2$. Then we study the asymptotic behaviour of of $u_{\epsilon_1,\epsilon_2}$ as $\epsilon_1$ and $\epsilon_2$ tend to zero. We show that the first two terms of an asymptotic approximation can be computed only if we introduce a suitable relation between $\epsilon_1$ and $\epsilon_2$

Singular behavior for a multi-parameter periodic Dirichlet problem

We consider a Dirichlet problem for the Poisson equation in a periodically perforated domain. The geometry of the domain is controlled by two parameters: a real number $\epsilon>0$ proportional to the radius of the holes and a map $\phi$, which models the shape of the holes. So, if $g$ denotes the Dirichlet boundary datum and $f$ the Poisson datum, we have a solution for each quadruple $(\epsilon,\phi,g,f)$. Our aim is to study how the solution depends on $(\epsilon,\phi,g,f)$, especially when $\epsilon$ is very small and the holes narrow to points. In contrast with previous works, we don't introduce the assumption that $f$ has zero integral on the fundamental periodicity cell. This brings in a certain singular behavior for $\epsilon$ close to $0$. We show that, when the dimension $n$ of the ambient space is greater than or equal to $3$, a suitable restriction of the solution can be represented with an analytic map of the quadruple $(\epsilon,\phi,g,f)$ multiplied by the factor $1/\epsilon^{n-2}$. In case of dimension $n=2$, we have to add $\log \epsilon$ times the integral of $f/2\pi$

Design and Development of a Planetary Gearbox for Electromechanical Actuator Test Bench through Additive Manufacturing

The development and validation of prognostic algorithms and digital twins for Electromechanical Actuators (EMAs) requires datasets of operating parameters that are not commonly available. In this context, we are assembling a test bench able to simulate different operating scenarios and environmental conditions for an EMA in order to collect the operating parameters of the actuator both in nominal conditions and under the effect of incipient progressive faults. This paper presents the design and manufacturing of a planetary gearbox for the EMA test bench. Mechanical components were conceived making extensive use of Fused Deposition Modelling (FDM) additive manufacturing and off-the-shelf hardware in order to limit the costs and time involved in prototyping. Given the poor mechanical properties of the materials commonly employed for FDM, the gears were not sized for the maximum torque of the electric motor, and a secondary torque path was placed in parallel of the planetary gearbox to load the motor through a disc brake. The architecture of the gearbox allowed a high gear ratio within a small form factor, and a bearingless construction with a very low number of moving parts

Dipendenza reale analitica da perturbazioni del supporto e della densità dei potenziali elastici di semplice e doppio strato

Il proposito di questa Tesi e' di studiare la dipendenza dei potenziali elastici di semplice e doppio strato dall'ipersuperfice di integrazione, cioe' dal supporto.openCorso di Laurea in matematic

Modelling of a safety relief valve through a MATLAB-Simulink and CFD based approach

The aim of the work is to understand the proper way to address the design and optimization procedures of a hydraulic safety relief valve. These valves are a part of the hydraulic circuit of many aircraft models, so their performances must be adapted to the specific system or engine. The only real constraints are the geometrical dimensions and the need to limit the weight of the device. This work requires gathering all the possible information available in the literature, and condensing them in a set of operations that will allow to promptly manufacture a product fitting the requirements needed. This should lead to the reduction of the amount of physical prototypes needed to obtain testing devices. The process studied uses a numerical fluid dynamic calculation approach to define the pressure field inside the valve and the forces acting on it, together with a Computational Fluid Dynamic (CFD) calculation used to identify the force distribution inside the valve. The first step deals with the creation of a CAD model of the valve. Then the CAD is imported into the CFD software, which evaluates the pressure field required to calculate the forces acting on the poppet of the valve. After the numerical scheme has been calibrated, some investigations are done to reduce the computational cost: the final goal is to run a complete simulation (meshing and solving) on a standard (even if high-end) laptop or desktop PC. Some of the positions (i.e. strokes) of the valve have been simulated as static, so a steady-state condition has been applied to solve the motion field. The main result consists of creating a MATLAB-Simulink® model capable to reach results comparable to that obtained by the CFD simulation, but in faster times. This means relying on a first-guess instrument, capable to address an initial design geometry. The further use of the Look-Up Tables (LUTs) increases the time required to obtain a solution, but links the Simulink model to the CFD simulation in order to reduce the amount of modeled quantities in favor of a greater precision of the model

Multi-parameter perturbations for the space periodic heat equation

This paper is divided into three parts. The first part focuses on periodic layer heat potentials, demonstrating their smooth dependence on regular perturbations of the support of integration. In the second part, we present an application of the results from the first part. Specifically, we consider a transmission problem for the heat equation in a periodic two-phase composite material and we show that the solution depends smoothly on the shape of the transmission interface, boundary data, and conductivity parameters. Finally, in the last part of the paper, we fix all parameters except for the contrast parameter and outline a strategy to deduce an explicit expansion of the solution using a Neumann-type series

Converging Expansions for Lipschitz Self-Similar Perforations of a Plane Sector

International audienceIn contrast with the well-known methods of matching asymptotics and multiscale (or compound) asymptotics, the " functional analytic approach " of Lanza de Cristoforis (Analysis 28, 2008) allows to prove convergence of expansions around interior small holes of size $ε$ for solutions of elliptic boundary value problems. Using the method of layer potentials, the asymptotic behavior of the solution as $ε$ tends to zero is described not only by asymptotic series in powers of $ε$, but by convergent power series. Here we use this method to investigate the Dirichlet problem for the Laplace operator where holes are collapsing at a polygonal corner of opening $ω$. Then in addition to the scale $ε$ there appears the scale $η = ε^{π/ω}$. We prove that when $π/ω$ is irrational, the solution of the Dirichlet problem is given by convergent series in powers of these two small parameters. Due to interference of the two scales, this convergence is obtained, in full generality, by grouping together integer powers of the two scales that are very close to each other. Nevertheless, there exists a dense subset of openings $ω$ (characterized by Diophantine approximation properties), for which real analyticity in the two variables $ε$ and $η$ holds and the power series converge unconditionally. When $π/ω$ is rational, the series are unconditionally convergent, but contain terms in log $ε$