Papers by Valeria Chiado' Piat
Applicable Analysis, 2018
We consider the asymptotic behaviour of integral energies with convex integrands defined on one-d... more We consider the asymptotic behaviour of integral energies with convex integrands defined on one-dimensional networks contained in a region of the three-dimensional space with a fast-oscillating boundary as the period of the oscillation tends to zero, keeping the oscillation themselves of fixed size. The limit energy, obtained as a Γlimit with respect to an appropriate convergence, is defined in a 'stratified' Sobolev space and is written as an integral functional depending on all, two or just one derivative, depending on the connectedness properties of the sublevels of the function describing the profile of the oscillations. In the three cases, the energy function is characterized through an usual homogenization formula for p-connected networks, a homogenization formula for thin-film networks and a homogenization formula for thin-rod networks, respectively.
Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 1990
Networks and Heterogeneous Media, 2012
We consider homogenization of Steklov spectral problem for a divergence form elliptic operator in... more We consider homogenization of Steklov spectral problem for a divergence form elliptic operator in periodically perforated domain under the assumption that the spectral weight function changes sign. We show that the limit behaviour of the spectrum depends essentially on wether the average of the weight function over the boundary of holes is positive, or negative or equal to zero. In all these cases we construct the asymptotics of the eigenpairs.
Lecture Notes of the Unione Matematica Italiana, 2006
Problems with multiple scales.- From discrete systems to continuous variational problems: an intr... more Problems with multiple scales.- From discrete systems to continuous variational problems: an introduction.- Relaxation for bulk and interfacial energies.- Convergence of Dirichlet forms on fractals.- Homogenization in perforated domains.- Homogenization of random non stationary parabolic operators.- Problems with concentration.- ?-convergence for concentration problems.- Gamma-convergence of gradient flows and applications to Ginzburg-Landau vortex dynamics.- PDE analysis of concentrating energies for the Ginzburg-Landau equation.
Networks & Heterogeneous Media, 2008
We prove a homogenization theorem for non-convex functionals depending on vector-valued functions... more We prove a homogenization theorem for non-convex functionals depending on vector-valued functions, defined on Sobolev spaces with respect to oscillating measures. The proof combines the use of the localization methods of Γ-convergence with a 'discretization' argument, which allows to link the oscillating energies to functionals defined on a single Lebesgue space, and to state the hypothesis of p-connectedness of the underlying periodic measure in a handy way.
SIAM Journal on Mathematical Analysis, 2013
This paper is aimed at homogenization of an elliptic spectral problem stated in a perforated doma... more This paper is aimed at homogenization of an elliptic spectral problem stated in a perforated domain, Fourier boundary conditions being imposed on the boundary of perforation. The presence of a locally periodic coefficient in the boundary operator gives rise to the effect of localization of the eigenfunctions. Moreover, the limit behavior of the lower part of the spectrum can be described in terms of an auxiliary harmonic oscillator operator. We describe the asymptotics of the eigenpairs and derive estimates for the rate of convergence.
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2003
We study the behaviour of non-convex functionals singularly perturbed by a possibly oscillating i... more We study the behaviour of non-convex functionals singularly perturbed by a possibly oscillating inhomogeneous gradient term, in the spirit of the gradient theory of phase transitions. We show that a limit problem giving a sharp interface, as the perturbation vanishes, always exists, but may be inhomogeneous or anisotropic. We specialize this study when the perturbation oscillates periodically, highlighting three types of regimes, depending on the frequency of the oscillations. In the two extreme cases, a separation of scales effect is described.
Physica B: Condensed Matter, 2012
This paper is devoted to the determination of the equivalent anisotropy properties of polycrystal... more This paper is devoted to the determination of the equivalent anisotropy properties of polycrystalline magnetic materials, modelled by an assembly of monocrystalline grains with a stochastic spatial distribution of easy axes. The mathematical theory of Γ-convergence is applied to homogenize the anisotropic term in the Gibbs free energy. The procedure is validated focusing on the micromagnetic computation of reversal processes in polycrystalline magnetic thin films.
Nonlinear Analysis: Theory, Methods & Applications, 1992
Manuscripta Mathematica, 1997
In this paper we prove the Hhlder continuity of local minimizers of integral functionals whose mo... more In this paper we prove the Hhlder continuity of local minimizers of integral functionals whose model is = IDul <~) dz, where ~ is an open subset ofIR '~, a E Wl'S(~2), s > n, a > 1 in Q, and u E W~o~(~2) is a scalar-valued function. Following the method introduced by De Giorgi in [6], the proof of the main result is based on suitable Caccioppoli and Sobolev-Poincar~ inequalities and on a fine estimate of the supremum of the local minimizers over small balls.
Manuscripta Mathematica, 1990
Page 1. manuscripta math. 68, 229 - 247 (1990) manuscripta mathema ti ca ~) Springer-Verlag 1990 ... more Page 1. manuscripta math. 68, 229 - 247 (1990) manuscripta mathema ti ca ~) Springer-Verlag 1990 HOMOGENIZATION OF QUASI-LINEAR EQUATIONS W~H NATURAL GROWTH TERMS Valcria CHIAD0 PIAT Annclicse DEFRANCESCHI Abstract ...
Journal of Magnetism and Magnetic Materials, 2008
This paper proposes a novel numerical procedure to evaluate the shielding factor of ferromagnetic... more This paper proposes a novel numerical procedure to evaluate the shielding factor of ferromagnetic grid shields, coupling the thin-shell formulation with the multiple scale expansion homogenization method. This approach is applied to three typical configurations for magnetic field mitigation under dc and 50Hz sinusoidal supply conditions, considering nickel and iron alloys. The influence of the shield hole dimension is finally
Journal of Differential Equations, 1994
Revista de la Real Academia de …, 2003
In this paper, we consider a family of scattering problems in perforated unbounded domains Ωε. We... more In this paper, we consider a family of scattering problems in perforated unbounded domains Ωε. We assume that the perforation is contained in a bounded region and that the holes have a 'critical' size. We study the asymptotic behaviour of the outgoing solutions of the steady-state scattering problem and we prove that an extra term appears in the limit equation. Finally, we obtain convergence results for scattering frequencies and solutions. Problemas de difracción en un dominio con pequeños agujeros. Resumen. En este artículo consideramos una familia de problemas de difracción en un dominio Ωε no limitado y perforado. Suponemos que las perforaciones están contenidas en una región limitada y que los agujeros tengan una talla crítica. Estudiamos el comportamiento asintótico de las soluciones que emergen del problema estacionario de difracción y probamos que en la ecuación límite, aparece un término nuevo. Finalmente, obtenemos algunos resultados de convergencia para las frecuencias y las soluciones de difracción.
IEEE Transactions on Magnetics, 2008
This paper presents a mathematical homogenization technique able to handle fine periodic structur... more This paper presents a mathematical homogenization technique able to handle fine periodic structures in presence of magnetic saturable/hysteretic media. The modeling approach, based on the multiple scale expansion theory, enables the evaluation both of equivalent electric parameters and effective magnetization curves (or hysteresis loops). The results of the homogenization technique are validated by comparison with those provided by a standard finite element solution of a two-dimensional current driven problem defined on the heterogeneous domain.
IEEE Transactions on Magnetics, 2005
A mathematical homogenization technique is applied to the computation of eddy currents in strip-w... more A mathematical homogenization technique is applied to the computation of eddy currents in strip-wound amorphous cores. The results are compared with a standard finite-element (FE) solution of a test problem. The method is applied to the analysis of an amorphous core in a frequency range up to 1 MHz, with main emphasis on energy loss prediction.
The European Physical Journal Applied Physics, 2007
This paper presents the application of a homogenisation technique, based on the multi-scale expan... more This paper presents the application of a homogenisation technique, based on the multi-scale expansion theory, to the analysis of heterogeneous materials constituted of magnetic inclusions dispersed in a dielectric lattice. The role of the shape and dimensions of the inclusions is analysed with reference to the effective electromagnetic properties and energy losses. The investigation is extended to the influence of flux waveforms with harmonic distortion, focusing the attention on the energy loss dependency on the harmonic content.
ESAIM: Control, Optimisation and Calculus of Variations, 2008
The work focuses on the Γ-convergence problem and the convergence of minimizers for a functional ... more The work focuses on the Γ-convergence problem and the convergence of minimizers for a functional defined in a periodic perforated medium and combining the bulk (volume distributed) energy and the surface energy distributed on the perforation boundary. It is assumed that the mean value of surface energy at each level set of test function is equal to zero. Under natural coercivity and p-growth assumptions on the bulk energy, and the assumption that the surface energy satisfies p-growth upper bound, we show that the studied functional has a nontrivial Γ-limit and the corresponding variational problem admits homogenization.
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Papers by Valeria Chiado' Piat