On asymptotic analysis of spectral problems in elasticity

Sokołowski, Jan

doi:10.1590/s1679-78252011000100003

Cited by 8 publications

(2 citation statements)

References 23 publications

(55 reference statements)

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“…The case of elasticity is also considered there, along with the extension to the scalar case with multiple defects. Asymptotic analysis of the spectral problem for elasticity in an anisotropic and inhomogeneous body has been carried out in [29]. The spectral problem for the plate containing a single small clamped hole and corresponding asymptotics of the first eigenvalue and corresponding eigenfunction can be found in [3].…”

Section: Highlights Of the Resultsmentioning

confidence: 99%

Eigenvalue Problem in a Solid with Many Inclusions: Asymptotic Analysis

Maz′ya¹,

Movchan²

2017

Multiscale Model. Simul.

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We construct the asymptotic approximation to the first eigenvalue and corresponding eigensolution of Laplace's operator inside a domain containing a cloud of small rigid inclusions. The separation of the small inclusions is characterised by a small parameter which is much larger compared with the nominal size of inclusions. Remainder estimates for the approximations to the first eigenvalue and associated eigenfield are presented. Numerical illustrations are given to demonstrate the efficiency of the asymptotic approach compared to conventional numerical techniques, such as the finite element method, for three-dimensional solids containing clusters of small inclusions.

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Section: Highlights Of the Resultsmentioning

confidence: 99%

Eigenvalue Problem in a Solid with Many Inclusions: Asymptotic Analysis

Maz′ya¹,

Movchan²

2017

Multiscale Model. Simul.

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“…This relatively new concept was introduced in the fundamental paper [56] and has been successfully applied to many relevant fields such as shape and topology optimization [1,8,11,12,15,17,29,38,40,48,49,50,59], inverse problems [10,19,20,21,23,30,32,34,36,42], imaging processing [13,14,31,33,39], multiscale material design [9,26,27,28,52] and mechanical modeling including damage [2] and fracture [60] evolution phenomena. Regarding the theoretical development of the topological asymptotic analysis, see for instance [6,7,22,24,25,35,37,41,43,44,45,46,47,…”

Section: Introductionmentioning

confidence: 99%

Topological Derivative Method for Electrical Impedance Tomography Problems

Ferreira

Novotny

Sokołowski

2016

IAPGOS

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In the field of shape and topology optimization the new concept is the topological derivative of a given shape functional. The asymptotic analysis is applied in order to determine the topological derivative of shape functionals for elliptic problems. The topological derivative (TD) is a tool to measure the influence on the specific shape functional of insertion of small defect into a geometrical domain for the elliptic boundary value problem (BVP) under considerations. The domain with the small defect stands for perturbed domain by topological variations. This means that given the topological derivative, we have in hand the first order approximation with respect to the small parameter which governs the volume of the defect for the shape functional evaluated in the perturbed domain. TD is a function defined in the original (unperturbed) domain which can be evaluated from the knowledge of solutions to BVP in such a domain. This means that we can evaluate TD by solving only the BVP in the intact domain. One can consider the first and the second order topological derivatives as well, which furnish the approximation of the shape functional with better precision compared to the first order TD expansion in perturbed domain. In this work the topological derivative is applied in the context of Electrical Impedance Tomography (EIT). In particular, we are interested in reconstructing a number of anomalies embedded within a medium subject to a set of current fluxes, from measurements of the corresponding electrical potentials on its boundary. The basic idea consists in minimize a functional measuring the misfit between the boundary measurements and the electrical potentials obtained from the model with respect to a set of ball-shaped anomalies. The first and second order topological derivatives are used, leading to a non-iterative second order reconstruction algorithm. Finally, a numerical experiment is presented, showing that the resulting reconstruction algorithm is very robust with respect to noisy data. ZASTOSOWANIE METODY POCHODNEJ TOPOLOGICZNEJ W ELEKTRYCZNEJ TOMOGRAFII IMPEDANCYJNEJStreszczenie. W dziedzinie optymalizacji kształtu i topologii zaproponowano nową koncepcję pochodnej topologicznej danego funkcjonału kształtu. Zastosowano asymptotyczną analizę w celu określenia pochodnej topologicznej funkcjonału kształtu dla zagadnień eliptycznych. Pochodna Topologiczna -PT (ang. the topological derivative -TD) jest miarą wpływu wtrącenia w postaci małego defektu na funkcjonał kształtu w badanym obszarze dla eliptycznego zagadnienia brzegowego. Obszar z małym defektem traktowany jest jako obszar zaburzony przez zmiany topologii. Oznacza to, że dana pochodna topologiczna stanowi aproksymację pierwszego rzędu ze względu na mały parametr, który określa objętość defektu dla obliczanego funkcjonału kształtu w zaburzonym obszarze. PT jest funkcją zdefiniowaną w obszarze niezaburzonym, który może być wyznaczony na podstawie znajomości rozwiązania zagadnienia brzegowego w tym (niezaburzonym) obszarze. Oznacza to że PT może być w...

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