We study the optimization of three dimensional curved rods and of shells
under minimal regularity assumptions for the geometry. The results that we
establish concern the existence of optimal shapes and the sensitivity analysis.
We also compute several numerical examples for the curved rods. The models
that we use have been investigated in our previous work [11], [16] and a
complete study of the Kirchhoff-Love arches and their optimization has been
performed in [10].
In this paper we propose a time discretization of a system of two parabolic equations describing diffusion-driven atom rearrangement in crystalline matter. The equations express the balances of microforces and microenergy; the two phase fields are the order parameter and the chemical potential. The initial and boundary-value problem for the evolutionary system is known to be well posed. Convergence of the discrete scheme to the solution of the continuous problem is proved by a careful development of uniform estimates, by weak compactness and a suitable treatment of
nonlinearities. Moreover, for the difference of discrete
and continuous solutions we prove an error estimate of
order one with respect to the time step.
The present note deals with a nonstandard systems of differential equations describing a two-species phase segregation. This system naturally arises in the asymptotic analysis carried out recently by the same authors,
as the diffusion coefficient in the equation governing
the evolution of the order parameter tends to zero. In particular, an existence result has been proved for the limit system in a very general framework. On the contrary, uniqueness was shown by assuming a constant mobility coefficient. Here, we generalize this result and prove
a continuous dependence property in the case that the mobility coefficient suitably depends on the chemical potential.
We are concerned with a nonstandard phase field model of
Cahn-Hilliard type. The model, which was introduced by Podio-Guidugli (Ric. Mat. 2006), describes two-species phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been recently investigated
by Colli, Gilardi, Podio-Guidugli, and Sprekels in a series of papers: see, in particular, SIAM J. Appl. Math. 2011,
and Boll. Unione Mat. Ital. 2012. In the latter contribution, the authors can treat the very general case in which the diffusivity coefficient of the parabolic PDE
is allowed to depend nonlinearly on both variables. In the same framework, this paper investigates the asymptotic limit of the solutions to the initial-boundary value problems as the diffusion coefficient sigma in the equation governing the evolution of the order parameter tends to zero. We prove that such a limit actually exists and solves the limit problem, which couples a nonlinear PDE of parabolic type with an ODE accounting for the phase dynamics. In the case of a constant diffusivity, we are able to show uniqueness and to improve the regularity of the solution.
We study a diffusion model of phase field type, consisting of
a system of two partial differential equations encoding the balances
of microforces and microenergy; the two unknowns are the order parameter
and the chemical potential. By a careful development of uniform estimates and
the deduction of certain useful boundedness properties, we
prove existence and uniqueness of a global-in-time
smooth solution to the associated initial/boundary-value problem;
moreover, we give a description of the relative $\omega$-limit set.
This paper is concerned with a diffusion model of phase-field type, consisting
of a {parabolic} system of two partial differential equations{,} interpreted as balances
of microforces and microenergy{, for two unknowns: the problem's order parameter $\rho$}
and the chemical potential $\mu$; each equation includes a viscosity term -- respectively, $\varepsilon \,\partial_t\mu$ and $\delta\,\partial_t\rho$ -- with $\varepsilon$ and $\delta$ two positive parameters; the field equations are complemented by Neumann homogeneous boundary conditions and suitable initial conditions. In a recent paper \cite{CGPS3}, we proved that this problem is \wepo\ and investigated the \loti\ \bhv\ of its $(\varepsilon,\delta)-$solutions. Here we discuss the asymptotic limit of the system as $\eps$
tends to $0$. We prove convergence of
$(\varepsilon,\delta)-$solutions to the corresponding solutions for
the case $\eps =0$, whose long-time behavior we characterize; in the
proofs, we employ compactness and monotonicity arguments.
We investigate a distributed optimal control problem for a phase field
model of Cahn-Hilliard type. The model describes two-species phase segregation
on an atomic lattice under the presence of diffusion; it has been introduced recently in
[4], on the basis of the theory developed in [15], and consists of a system of two
highly nonlinearly coupled PDEs. For this reason, standard arguments of optimal control theory do not apply
directly, although the control constraints and the cost functional are of standard type.
We show that the problem admits a solution, and we derive the first-order
necessary conditions of optimality.
Global existence and uniqueness for a singular/degenerate Cahn-Hilliard system with viscosity
(2013)
Existence and uniqueness are investigated for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system aims to model two-species phase segregation on an atomic lattice [19]; in the balance equations of microforces and microenergy, the two
unknowns are the order parameter rho and the chemical potential mu. A simpler version of the same system has recently been discussed in [8]. In this paper, a fairly more general phase-field equation for rho is coupled with a genuinely nonlinear diffusion equation for mu. The existence of a global-in-time solution is proved with the help of suitable a priori estimates. In the case of constant atom mobility, a new and rather unusual uniqueness
proof is given, based on a suitable combination of variables.
This note is concerned with a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. The system arises from a model of two-species phase segregation
on an atomic lattice [22]; it consists of the balance equations of microforces and microenergy; the two unknowns are the order parameter rho and the chemical potential mu. Some recent results obtained for this class of problems is reviewed and, in the case of a nonconstant and nonlinear atom mobility, uniqueness and continuous dependence on the initial data are shown with the help of a new line of
argumentation developed in [12].