Para el estudio de la naturaleza de formas críticas en optimización de formas se requieren algunas propiedades de continuidad sobre las derivadas de segundo orden de las formas. Dado que la fórmula de Taylor-Young involucra a diferentes topologías que no son equivalentes, dicha fórmula no permite deducir cuando una forma crítica es un mínimo local estricto de la función forma pese a que su Hessiano sea definido positivo en ese punto. El resultado principal de este trabajo ofrece una cota superior...
This work deals with a two-dimensional inverse problem in the field of tomography. The geometry of an unknown inclusion has to be reconstructed from boundary measurements. In this paper, we extend previous results of R. Kress and his coauthors: the leading idea is to use the conformal mapping function as unknown. We establish an integrodifferential equation that the trace of the Riemann map solves. We write it as a fixed point equation and give conditions for contraction. We conclude with a series...
In this work, we consider singular perturbations of the boundary of a smooth domain. We describe the asymptotic behavior of the solution u of a second order elliptic equation posed in the perturbed domain with respect to the size parameter of the deformation. We are also interested in the variations of the energy functional. We propose a numerical method for the approximation of u based on a multiscale superposition of the unperturbed solution
and a profile defined in a model...
The level set method has become widely used in shape optimization where it allows a popular implementation of the steepest descent method. Once coupled with a ersatz material approximation [Allaire ,
(2004) 363–393], a single mesh is only used leading to very efficient and cheap numerical schemes in optimization of structures. However, it has some limitations and cannot be applied in every situation. This work aims at exploring such a limitation. We estimate the systematic error...
We discuss the stability of "critical" or "equilibrium" shapes of
a shape-dependent energy functional. We analyze a problem arising when
looking at the positivity of the second derivative in order to prove
that a critical shape is an optimal shape. Indeed, often when
positivity -or coercivity- holds, it does for a weaker norm than the
norm for which the functional is twice differentiable and local
optimality cannot be deduced. We solve this problem for a
particular but significant example. We...
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