2,641 research outputs found
Shape Minimization of Dendritic Attenuation
What is the optimal shape of a dendrite? Of course, optimality refers to some
particular criterion. In this paper, we look at the case of a dendrite sealed
at one end and connected at the other end to a soma. The electrical potential
in the fiber follows the classical cable equations as established by W. Rall.
We are interested in the shape of the dendrite which minimizes either the
attenuation in time of the potential or the attenuation in space. In both
cases, we prove that the cylindrical shape is optimal
On a Bernoulli problem with geometric constraints
A Bernoulli free boundary problem with geometrical constraints is studied.
The domain \Om is constrained to lie in the half space determined by and its boundary to contain a segment of the hyperplane where
non-homogeneous Dirichlet conditions are imposed. We are then looking for the
solution of a partial differential equation satisfying a Dirichlet and a
Neumann boundary condition simultaneously on the free boundary. The existence
and uniqueness of a solution have already been addressed and this paper is
devoted first to the study of geometric and asymptotic properties of the
solution and then to the numerical treatment of the problem using a shape
optimization formulation. The major difficulty and originality of this paper
lies in the treatment of the geometric constraints
On the controllability of quantum transport in an electronic nanostructure
We investigate the controllability of quantum electrons trapped in a
two-dimensional device, typically a MOS field-effect transistor. The problem is
modeled by the Schr\"odinger equation in a bounded domain coupled to the
Poisson equation for the electrical potential. The controller acts on the
system through the boundary condition on the potential, on a part of the
boundary modeling the gate. We prove that, generically with respect to the
shape of the domain and boundary conditions on the gate, the device is
controllable. We also consider control properties of a more realistic nonlinear
version of the device, taking into account the self-consistent electrostatic
Poisson potential
What is the optimal shape of a fin for one dimensional heat conduction?
This article is concerned with the shape of small devices used to control the
heat flowing between a solid and a fluid phase, usually called \textsl{fin}.
The temperature along a fin in stationary regime is modeled by a
one-dimensional Sturm-Liouville equation whose coefficients strongly depend on
its geometrical features. We are interested in the following issue: is there
any optimal shape maximizing the heat flux at the inlet of the fin? Two
relevant constraints are examined, by imposing either its volume or its
surface, and analytical nonexistence results are proved for both problems.
Furthermore, using specific perturbations, we explicitly compute the optimal
values and construct maximizing sequences. We show in particular that the
optimal heat flux at the inlet is infinite in the first case and finite in the
second one. Finally, we provide several extensions of these results for more
general models of heat conduction, as well as several numerical illustrations
Spectral shape optimization for the Neumann traces of the Dirichlet-Laplacian eigenfunctions
We consider a spectral optimal design problem involving the Neumann traces of
the Dirichlet-Laplacian eigenfunctions on a smooth bounded open subset
of . The cost functional measures the amount of energy that Dirichlet
eigenfunctions concentrate on the boundary and that can be recovered with a
bounded density function. We first prove that, assuming a constraint on
densities, the so-called {\it Rellich functions} maximize this
functional.Motivated by several issues in shape optimization or observation
theory where it is relevant to deal with bounded densities, and noticing that
the -norm of {\it Rellich functions} may be large, depending on the
shape of , we analyze the effect of adding pointwise constraints when
maximizing the same functional. We investigate the optimality of {\it
bang-bang} functions and {\it Rellich densities} for this problem. We also deal
with similar issues for a close problem, where the cost functional is replaced
by a spectral approximation.Finally, this study is completed by the
investigation of particular geometries and is illustrated by several numerical
simulations
Optimal shape and location of sensors for parabolic equations with random initial data
In this article, we consider parabolic equations on a bounded open connected
subset of . We model and investigate the problem of optimal
shape and location of the observation domain having a prescribed measure. This
problem is motivated by the question of knowing how to shape and place sensors
in some domain in order to maximize the quality of the observation: for
instance, what is the optimal location and shape of a thermometer? We show that
it is relevant to consider a spectral optimal design problem corresponding to
an average of the classical observability inequality over random initial data,
where the unknown ranges over the set of all possible measurable subsets of
of fixed measure. We prove that, under appropriate sufficient spectral
assumptions, this optimal design problem has a unique solution, depending only
on a finite number of modes, and that the optimal domain is semi-analytic and
thus has a finite number of connected components. This result is in strong
contrast with hyperbolic conservative equations (wave and Schr\"odinger)
studied in [56] for which relaxation does occur. We also provide examples of
applications to anomalous diffusion or to the Stokes equations. In the case
where the underlying operator is any positive (possible fractional) power of
the negative of the Dirichlet-Laplacian, we show that, surprisingly enough, the
complexity of the optimal domain may strongly depend on both the geometry of
the domain and on the positive power. The results are illustrated with several
numerical simulations
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