9,794 research outputs found
Optimal actuator design based on shape calculus
An approach to optimal actuator design based on shape and topology
optimisation techniques is presented. For linear diffusion equations, two
scenarios are considered. For the first one, best actuators are determined
depending on a given initial condition. In the second scenario, optimal
actuators are determined based on all initial conditions not exceeding a chosen
norm. Shape and topological sensitivities of these cost functionals are
determined. A numerical algorithm for optimal actuator design based on the
sensitivities and a level-set method is presented. Numerical results support
the proposed methodology.Comment: 41 pages, several figure
Maximization of Laplace-Beltrami eigenvalues on closed Riemannian surfaces
Let be a connected, closed, orientable Riemannian surface and denote
by the -th eigenvalue of the Laplace-Beltrami operator on
. In this paper, we consider the mapping .
We propose a computational method for finding the conformal spectrum
, which is defined by the eigenvalue optimization problem
of maximizing for fixed as varies within a conformal
class of fixed volume . We also propose a
computational method for the problem where is additionally allowed to vary
over surfaces with fixed genus, . This is known as the topological
spectrum for genus and denoted by . Our
computations support a conjecture of N. Nadirashvili (2002) that
, attained by a sequence of surfaces degenerating to
a union of identical round spheres. Furthermore, based on our computations,
we conjecture that ,
attained by a sequence of surfaces degenerating into a union of an equilateral
flat torus and identical round spheres. The values are compared to
several surfaces where the Laplace-Beltrami eigenvalues are well-known,
including spheres, flat tori, and embedded tori. In particular, we show that
among flat tori of volume one, the -th Laplace-Beltrami eigenvalue has a
local maximum with value . Several properties are also studied
computationally, including uniqueness, symmetry, and eigenvalue multiplicity.Comment: 43 pages, 18 figure
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