PL
 |
EN
Preferencje help
Polish version English version Język
Widoczny [Schowaj] Abstrakt
Liczba wyników

Znaleziono wyników: 12

Liczba wyników na stronie
first rewind previous Strona / 1 next fast forward last
Wyniki wyszukiwania
Wyszukiwano:
w słowach kluczowych:  pochodna topologiczna
help Sortuj według:

help Ogranicz wyniki do:
first rewind previous Strona / 1 next fast forward last
1
Content available remote Minimization of objective function in electrical impedance tomography by topological derivative
Rymarczyk Tomasz, Szulc Katarzyna
Przegląd Elektrotechniczny
|
2019
|
R. 95, nr 6
137--140
EN
The article presents the reconstruction of 2D objects studied using the topological derivative and level set function in electrical impedance tomography, which is a non-invasive imaging method in which an unknown physical object is examined using measurements on its edge. The internal distribution of conductivity is obtained on the basis of the measurements. The solution to the optimization problem is obtained by combining finite element methods and topological algorithms. The presented solution can be effectively used in applications based on electrical tomography.
PL
W artykule przedstawiono rekonstrukcję badanych obiektów 2D z wykorzystaniem pochodnej topologicznej i funkcji zbiorów poziomicowych w elektrycznej tomografii impedancyjnej, która jest nieinwazyjną metodą obrazowania, w której nieznany obiekt fizyczny jest badany za pomocą pomiarów na jego krawędzi. Wewnętrzny rozkład konduktywności jest otrzymywany na podstawie pomiarów. Rozwiązanie problemu optymalizacji uzyskuje się przez połączenie metody elementów skończonych i algorytmów topologicznych. Prezentowane rozwiązanie może być skutecznie wykorzystywane w aplikacjach opartych na tomografii elektrycznej.
2
Content available A design of an optimal shape of domain described by NURBS curves using the topological derivative and boundary element method
Freus K., Freus S.
Journal of Applied Mathematics and Computational Mechanics
|
2017
|
Vol. 16, nr 2
67--76
EN
The aim of this paper is to create an optimal shape of the 2D domain that is described by the Non-Uniform Rational B-Splines (NURBS) curves. This work presents a method based on the topological derivative for the Laplace equation that determines the sensitivity of a given cost function to the change of its topology. As a numerical approach, the boundary element method is considered. To check the effectiveness of the proposed approach, the example of computations was carried out.
3
Content available Topological derivative method for electrical impedance tomography problems
Ferreira A. D., Novotny A. A., Sokołowski J.
Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska
|
2016
|
nr 2
4--8
EN
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.
PL
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 wyznaczona poprzez rozwiązanie zagadnienia brzegowego w obszarze niezaburzonym. Można rozważyć pierwszego jak również drugiego rzędu pochodną topologiczną, zapewniającą aproksymację funkcjonału kształtu ze znacznie lepszą precyzją w porównaniu do PT pierwszego rzędu rozwinięcia w obszarze zaburzonym. W niniejszej pracy PT jest zastosowana w kontek- ście Elektrycznej Tomografii Impedancyjnej (ETI). W szczególności jesteśmy zainteresowani w rekonstrukcji pewnej liczby anomalii wewnątrz obszaru, na podstawie pomiarów potencjału na brzegu rozpatrywanego obszaru. Podstawowa idea zawarta jest w minimalizacji funkcjonału, będącego miarą niedopasowania między pomiarami potencjału na brzegu obszaru a potencjałem elektrycznym uzyskanym na podstawie modelu matematycznego uwzględniającego zbiór anomalii o kształcie kuli. Zastosowanie pierwszego i drugiego rzędu pochodnej topologicznej prowadzi do nieiteracyjnego algorytmu rekonstrukcyjnego drugiego rzędu. W zakończeniu artykułu przedstawiono eksperyment numeryczny, wykazujący, że zaproponowany algorytm obrazowania jest bardzo odporny na zaszumione dane pomiarowe.
4
Content available Topological derivative - theory and applications
Szulc K.
Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska
|
2015
|
nr 1
7--13
EN
The paper is devoted to present some mathematical aspects of the topological derivative and its applications in different fields of sciences such as shape optimization and inverse problems. First the definition of the topological derivative is given and the shape optimization problem is formulated. Next the form of the topological derivative is evaluated for a mixed boundary value problem defined in a geometrical domain. Finally, an example of an application of the topological derivative in the electric impedance tomography is presented.
PL
W pracy przedstawiono matematyczne aspekty dotyczące pochodnej topologicznej oraz jej zastosowań w różnych dziedzinach nauki, takich jak optymalizacja kształtu czy problemy odwrotne. W pierwszej części podano nieformalna˛ definicje˛ pochodnej topologicznej oraz sformułowano problem optymalizacji kształtu. Następnie wyprowadzono postać pochodnej topologicznej dla mieszanego problemu brzegowego. W ostatniej części przedstawiono przykład zastosowania pochodnej topologicznej dla problemu elektrycznej tomografii impedancyjnej.
5
Content available Determination of an optimal shape of domain using the topological derivative and the Boundary Element Method
Freus K., Freus S.
Journal of Applied Mathematics and Computational Mechanics
|
2014
|
Vol. 13, nr 4
41--48
EN
In the paper, the topological derivative for the Laplace equation is taken into account. The governing equation is solved by means of the Boundary Element Method. The topological-shape sensitivity method is used to determine the points showing the lowest sensitivities. On the selected points, material is eliminated by opening a hole, using the appropriate iterative process. This one is halted when a given amount of material is removed. The objective of this work is to obtain an optimal topology of the domain considered. In the final part of the paper, the example of computations is shown.
6
Content available remote Topological derivatives for semilinear elliptic equations
Iguernane M., Nazarov S. A., Roche J. R., Sokolowski J., Szulc K.
International Journal of Applied Mathematics and Computer Science
|
2009
|
Vol. 19, no 2
191-205
EN
The form of topological derivatives for an integral shape functional is derived for a class of semilinear elliptic equations. The convergence of finite element approximation for the topological derivatives is shown and the error estimates in the L [...] norm are obtained. The results of numerical experiments which confirm the theoretical convergence rate are presented.
7
Content available remote Optimal internal dissipation of a damped wave equation using a topological approach
Münch A
International Journal of Applied Mathematics and Computer Science
|
2009
|
Vol. 19, no 1
15-37
EN
We consider a linear damped wave equation defined on a two-dimensional domain [...], with a dissipative term localized in a subset [...]. We address the shape design problem which consists in optimizing the shape of [...] in order to minimize the energy of the system at a given time T. By introducing an adjoint problem, we first obtain explicitly the (shape) derivative of the energy at time T with respect to the variation in [...]. Expressed as a boundary integral on [...], this derivative is then used as an advection velocity in a Hamilton-Jacobi equation for shape changes. We use the level-set methodology on a fixed working Eulerian mesh as well as the notion of the topological derivative. We also consider optimization with respect to the value of the damping parameter. The numerical approximation is presented in detail and several numerical experiments are performed which relate the over-damping phenomenon to the well-posedness of the problem.
8
Content available remote A level set method in shape and topology optimization for variational inequalities
Fulmański P., Laurain A., Scheid J. F., Sokołowski J.
International Journal of Applied Mathematics and Computer Science
|
2007
|
Vol. 17, no 3
413-430
EN
The level set method is used for shape optimization of the energy functional for the Signorini problem. The boundary variations technique is used in order to derive the shape gradients of the energy functional. The conical differentiability of solutions with respect to the boundary variations is exploited. The topology modifications during the optimization process are identified by means of an asymptotic analysis. The topological derivatives of the energy shape functional are employed for the topology variations in the form of small holes. The derivation of topological derivatives is performed within the framework proposed in (Sokołowski and Żochowski, 2003). Numerical results confirm that the method is efficient and gives better results compared with the classical shape optimization techniques.
9
Topological sensitivity derivative : application in optimal design and material science
Mróz Z., Bojczuk D.
Foundations of Civil and Environmental Engineering
|
2006
|
No. 7
229-259
EN
The concept of topologicai derivative is introduced and applied to optimal design of structural elements and to study the material microstructure evolution. For structural design the objective function and constraints provide the optimal design, for material microstructure the free energy and dissipation function generate the process of evolution such as phase transformation, crack growth or void generation. Three general modes of topology variation have been considered: generation of new elements, removing of the existing elements and a substitution of the existing elements by new elements. The cases of infinitesimal and finite topology variations have been discussed and illustrated by examples.
10
Content available remote Topological derivative for linear elastic plate bending problems
Novotny A. A., Feijoo R. A., Padra C., Taroco E.
Control and Cybernetics
|
2005
|
Vol. 34, no 1
339-361
EN
This study concerns the application of the Topological-Shape Sensitivity Method as a systematic procedure to determine the Topological Derivative for linear elastic plate bending problems within the framework of Kirchhoff's kinematic approach. This method, based on classical Shape Sensitivity Analysis, leads to a constructive procedure to obtain the Topological Derivative. Utilising the well known terminology of structural optimization, we adopt, the total potential strain energy as the cost function and the equilibrium equation as the constraint. Variational formulation as well as the direct differentiation method are used to perform the shape derivative of the cost function. Finally, in order to obtain a uniform distribution of bending moments in several plate problems, the Topological Derivative was approximated, by the Finite Element Method, and used to find the best place to insert holes. A simple hard-kill like topology algorithm, which furnishes satisfactory qualitative results in agreement with those reported in the literature, is also shown.
11
Content available remote Influence of a boundary perforation on the Dirichlet energy
Dambrine M., Vial G.
Control and Cybernetics
|
2005
|
Vol. 34, no 1
117-136
EN
We consider some singular perturbations of the boundary of a smooth domain. Such domain variations are not differentiable within the classical theory of shape calculus. We mimic the topological asymptotic and we derive an asymptotic expansion of the shape function in terms of a size parameter. The two-dimensional case of the Dirichlet energy is treated in detail. We give a full theoretical proof as well as a numerical confirmation of the results.
12
Content available remote A new approach for simultaneous shape and topology optimization based on dynamic implicit surface function
Guo H., Zhao K., Wang M. Y.
Control and Cybernetics
|
2005
|
Vol. 34, no 1
255-282
EN
In the present paper, a new approach for structural topology optimization based on dynamic implicit surface function (DISF) is proposed. DISF is used to describe the shape/topology of a structure, which is approximated in terms of the nodal values. Then, a relationship is established between the element stiffness and the values of the implicit surface function on its four nodes. In this way and with some non-local treatments of the design sensitivities, not only the shape derivative but also the topological derivative of the optimal design can be incorporated in the numerical algorithm in a unified way. Numerical experiments demonstrate that by employing this approach, the computational efforts associated with DISF (and level set) based algorithms can be diminished. Clear optimal topologies and smooth structural boundaries free from any sign of numerical instability can be obtained simultaneously and efficiently.
first rewind previous Strona / 1 next fast forward last
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.