An introduction to the topological asymptotic expansion with examples

Revue Africaine de la Recherche en Informatique et Mathématiques Appliquées ◽

10.46298/arima.1866 ◽

2006 ◽

Vol Volume 5, Special Issue TAM... ◽

Author(s):

Jérôme Fehrenbach ◽

Mohamed Masmoudi

Keyword(s):

Characteristic Function ◽

Small Area ◽

Optimal Shape ◽

Optimal Domain ◽

Petite Taille ◽

International Audience ◽

Gradient Type ◽

Type Information ◽

Topological Asymptotic Expansion ◽

Domain Of Validity

International audience To find an optimal domain is equivalent to look for Its characteristic function. At first sight this problem seems to be nondifferentiable. But it is possible to derive the variation of a cost function when we switch the characteristic function from 0 to 1 or from 1 to 0 a small area. Classical and two generalized adjoint approaches are considered in this paper. Their domain of validity is given and Illustrated by several examples. Using this gradient type Information, It is possible to build fast algorithms. Generally, only one Iteration Is needed to find the optimal shape. Trouver un domaine optimal est équivalent à la recherche de sa fonction caractéristique. A première vue, ce problème semble non différentiable, mais Il est possible de calculer la variation de la fonction coût lorsque la fonction caractéristique passe de 1 à 0 ou de 0 à 1 dans une région de petite taille. On s’appuiera sur une approche adjointe classique et deux généralisations de cette méthode. Le domaine de validité de ces différentes approches est donné et illustré par différents exemples. Cette Information de type gradient permet de construire des algorithmes très efficaces: en général, une seule Itération suffit pour trouver le domaine optimal.

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Topological asymptotic formula for the 3D non-stationary Stokes problem and application

Revue Africaine de la Recherche en Informatique et Mathématiques Appliquées ◽

10.46298/arima.4760 ◽

2020 ◽

Vol Volume 32 - 2019 - 2020 ◽

Author(s):

Maatoug Hassine ◽

Rakia Malek

Keyword(s):

Asymptotic Expansion ◽

Stokes System ◽

Stokes Problem ◽

Three Dimensional ◽

Detection Algorithm ◽

International Audience ◽

Flow Domain ◽

Topological Asymptotic Expansion ◽

Small Obstacle ◽

Theoretical Results

International audience This paper is concerned with a topological asymptotic expansion for a parabolic operator. We consider the three dimensional non-stationary Stokes system as a model problem and we derive a sensitivity analysis with respect to the creation of a small Dirich-let geometric perturbation. The established asymptotic expansion valid for a large class of shape functions. The proposed analysis is based on a preliminary estimate describing the velocity field perturbation caused by the presence of a small obstacle in the fluid flow domain. The obtained theoretical results are used to built a fast and accurate detection algorithm. Some numerical examples issued from a lake oxygenation problem show the efficiency of the proposed approach. Ce papier porte sur l'analyse de sensibilité topologique pour un opérateur parabolique. On considère le problème de Stokes instationnaire comme un exemple de modèle et on donne une étude de sensibilité décrivant le comportement asymptotique de l'opérateur relativement à une petite perturbation géométrique du domaine. L'analyse présentée est basée sur une estimation du champ de vitesse calculée dans le domaine perturbé. Les résultats de cette étude ont servi de base pour développer un algorithme d'identification géométrique. Pour la validation de notre approche, on donne une étude numérique pour un problème d'optimisation d'emplacement des injecteurs dans un lac eutrophe. Des exemples numériques montrent l'efficacité de la méthode proposée

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Existence of an optimal domain for the buckling load of a clamped plate with prescribed volume

Annali di Matematica Pura ed Applicata (1923 -) ◽

10.1007/s10231-021-01175-6 ◽

2021 ◽

Author(s):

Kathrin Stollenwerk

Keyword(s):

Free Boundary ◽

Boundary Value Problem ◽

Boundary Value ◽

Optimal Shape ◽

Buckling Load ◽

Volume Constraint ◽

Free Boundary Value Problem ◽

Optimal Domain ◽

Clamped Plate

AbstractWe formulate the minimization of the buckling load of a clamped plate as a free boundary value problem with a penalization term for the volume constraint. As the penalization parameter becomes small, we show that the optimal shape problem with prescribed volume is solved. In addition, we discuss two different choices for the penalization term.

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Shape optimization for the eigenfrequency of the plate

Georgian Mathematical Journal ◽

10.1515/gmj-2017-0005 ◽

2018 ◽

Vol 25 (1) ◽

pp. 19-24

Author(s):

Yusif S. Gasimov ◽

Natavan A. Allahverdiyeva

Keyword(s):

Optimal Shape ◽

Necessary Condition ◽

Biharmonic Operator ◽

Explicit Formulas ◽

Optimal Domain ◽

First Variation ◽

Transverse Vibrations ◽

The First Variation ◽

Domain Functional ◽

New Formula

AbstractIn this paper, we consider an eigenvalue problem for the biharmonic operator that describes the transverse vibrations of the plate. Under the imposed boundary conditions, the eigenvalues of this operator are indeed eigenfrequencies of the clamped plate. The domain of the plate is taken variable and the domain functional, involving an eigenfrequency, is studied. A new formula for an eigenfrequency is proved, the first variation of the functional with respect to the domain is calculated, and the necessary condition for an optimal shape is derived. New explicit formulas are obtained for the eigenfrequency in the optimal domain in some particular cases.

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Rocking Beam Microarea Diffraction Applied to Metallic thin Films

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100091962 ◽

1976 ◽

Vol 34 ◽

pp. 396-397

Author(s):

R. H. Geiss

Keyword(s):

Thin Films ◽

Electron Diffraction ◽

Small Area ◽

Objective Lens ◽

Electron Optics ◽

Metallic Thin Films ◽

Transmission Electron Diffraction ◽

Transmission Electron ◽

Sample Height ◽

Scanning Transmission

The theory and practical limitations of micro area scanning transmission electron diffraction (MASTED) will be presented. It has been demonstrated that MASTED patterns of metallic thin films from areas as small as 30 Åin diameter may be obtained with the standard STEM unit available for the Philips 301 TEM. The key to the successful application of MASTED to very small area diffraction is the proper use of the electron optics of the STEM unit. First the objective lens current must be adjusted such that the image of the C2 aperture is quasi-stationary under the action of the rocking beam (obtained with 40-80-160 SEM settings of the P301). Second, the sample must be elevated to coincide with the C2 aperture image and its image also be quasi-stationary. This sample height adjustment must be entirely mechanical after the objective lens current has been fixed in the first step.

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