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1

On the robustness of the topological derivative for Helmholtz problems and applications

Leugering Günter, Novotny Antonio André, Sokolowski Jan

Control and Cybernetics

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2022

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Vol. 51, No. 2

227--248

EN

We consider Helmholtz problems in two and three dimensions. The topological sensitivity of a given cost function J(uɛ) with respect to a small hole Bɛ around a given point x0ɛ ∈ Bɛ ⊂ Ω depends on various parameters, like the frequency k chosen or certain material parameters or even the shape parameters of the hole Bɛ. These parameters are either deliberately chosen in a certain range, as, e.g., the frequencies, or are known only up to some bounds. The problem arises as to whether one can obtain a uniform design using the topological gradient. We show that for 2-d and 3-d Helmholtz problems such a robust design is achievable.

2

Minimization of objective function in electrical impedance tomography by topological derivative

Rymarczyk Tomasz, Szulc Katarzyna

Przegląd Elektrotechniczny

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2019

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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.

3

Growth control of cracks under contact conditions based on the topological derivative of the Rice’s integral

da Silva Xavier Marcel Duarte, Novotny Antonio André, Sokołowski Jan

Control and Cybernetics

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2019

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Vol. 48, No. 2

307--323

EN

In the present paper we propose a simple method for dealing with growth control of cracks under contact type boundary conditions on their lips. The aim is to find a mechanism for decreasing the energy release rate of cracked components, which means increasing their fracture toughness. The method consists in minimizing a shape functional defined in terms of the Rice’s integral, with respect to the nucleation of hard and/or soft inclusions, according to the information provided by the associated topological derivative. Based on Griffith’s energy criterion, this simple strategy allows for an increase in fracture toughness of the cracked component. Since the problem is non-linear, the domain decomposition technique, combined with the Steklov-Poincaré pseudo-differential boundary operator, is used to obtain the sensitivity of the associated shape functional with respect to the nucleation of a small circular inclusion with different material property from the background. Then, the obtained topological derivatives are used to indicate the regions, where the controls should be positioned in order to solve the minimization problem we are dealing with. Finally, a numerical example is presented showing the applicability of the proposed methodology.

4

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

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2017

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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.

5

Topological sensitivity analysis for a coupled nonlinear problem with an obstacle

Abdelbari M., Nachi K., Sokolowski J.

Control and Cybernetics

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2017

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Vol. 46, no. 1

5--25

EN

The Topological Derivative has been recognized as a powerful tool in obtaining the optimal topology for several kinds of engineering problems. This derivative provides the sensitivity of the cost functional for a boundary value problem for nucleation of a small hole or a small inclusion at a given point of the domain of integration. In this paper, we present a topological asymptotic analysis with respect to the size of singular domain perturbation for a coupled nonlinear PDEs system with an obstacle on the boundary. The domain decomposition method, referring to the SteklovPoincar´epseudo-diﬀerential operator, is employed for the asymptotic study of boundary value problem with respect to the size of singular domain perturbation. The method is based on the observation that the known expansion of the energy functional in the ring coincides with the expansion of the Steklov-Poincar´e operator on the boundary of the truncated domain with respekt to the small parameter, which measures the size of perturbation. In this way, the singular perturbation of the domain is reduced to the regular perturbation of the Steklov-Poincar´e map ping for the ring. The topological derivative for a tracking type shape functional is evaluated so as to obtain the useful formula for application in the numerical methods of shape and topology optimization.

6

Applying the topological derivative and the boundary element method to design of shape of the heat exchanger

Freus K., Freus S.

Journal of Applied Mathematics and Computational Mechanics

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2015

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Vol. 14, nr 2

5--12

EN

In this work, the topological derivative for the Laplace equation is used to solve a design problem. This derivative describes the sensitivity of the problem when a small hole is formed at an arbitrary point of the domain. The goal of this work is to design topology of the domain when the Robin condition is imposed on the holes. Physically, the holes can be construed as cooling channels. For finding the solution of the governing equation the boundary element method is applied. The final part of the paper presents the design of the heat exchanger and results of computations.

7

Topological derivative - theory and applications

Szulc K.

Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska

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2015

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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.

8

Optimal design of bar structures with their supports in problems of stability and free vibrations

Bojczuk D., Rębosz-Kurdek A.

Journal of Theoretical and Applied Mechanics

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2014

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Vol. 52 nr 2

533--546

EN

The problem of maximization of the buckling load and the problem of maximization of the natural vibration frequency under a condition imposed on the global cost is discussed. Cross-sectional areas of bar structures and number of elastic supports, their positions and stiffnesses (or the number and positions of rigid supports) are selected as design parameters. The proposed here algorithm of optimization of bar structures with their supports is applied for analysis of some optimization problems. Illustrative examples confirm applicability of the proposed approach.

9

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

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2014

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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.

10

Shape and topology optimization of distributed parameter systems

Sokołowski J., Żochowski A.

Control and Cybernetics

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2013

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Vol. 42, no. 1

217--226

EN

The energy functional for an elliptic boundary value problem in two spatial dimensions is considered. The variations of shape functional resulting from the small shape-topological domain perturbations with the holes and inclusions in elastic body are determined. The exact representation of solutions to the boundary value problem is exploited for the purposes of asymptotic analysis. To this end the perturbed solutions of the boundary value problem are Expressem as the minimizers of perturbed energy functionals. The proposed method of asymptotic analysis results in the double asymptotic expansions, with respect to the size of a hole and to the contrast parameter of an inclusion with respect to the matrix, of solutions to the boundary value problems as well as of the associated energy functional. The shape sensitivity analysis of the energy functional with respekt of the boundary variations of an inclusion is performed. The further asymptotic analysis allows for the limit passage with the size of inclusion to zero. In this way the topological derivative of the energy functional is obtained. The proposed analysis can be used in the shape and topology optimum design for elastic bodies governed by the stationary as well as by the time dependent elasticity boundary value problems in the framework of selfadjoint extensions of elliptic operators.

11

Growth of solid bodies in the framework of shape and topology optimization

Ganghoffer J. -F., Sokolowski J.

Computer Methods in Materials Science

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2012

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Vol. 12, No. 1

9-19

EN

Ventricular assist device is an artificial organ, which is used to treat heart diseases. In the world, as well as in Poland, efforts are made towards the development of such a device that is biocompatible, durable, low energy consuming, allows monitoring and does not introduce changes to the blood morphology. The review paper discusses the types of ventricular assist devices (VADs), including VADs proposed in Poland. The particular emphasis is put on the numerical modelling and computer aided design of such an artificial organ. The walls of the ventricular assist device are covered with a nanocoating of TiN using modern techniques (Pulsed Laser Deposition) to improve the biocompatibility. The nanocoating modifies the surface properties of the device. Mechanical properties of nanocoating are determined in experimental nanotests and using imaging techniques of nanostructures. However, these tests give average values of properties and this information is not sufficient for advanced designof ventricular assist devices. To eliminate this constraint, the multiscale modelling is applied. Developed solution, which is based on application and combination of methods such as finite element method, multiscale approach and inverse analysis, is presented in the review paper. These methods are helpful in prediction the location of failure zones in the material of the ventricular assist device and then to analyze the local behaviour of nanocoating. Furthermore, it is possible to identify the parameters of the rheological model of nanocoating and introduce the residual stresses into models.

PL

W pracy wprowadza się nowe, quasi-statyczne ujęcie opisu wzrostu ciała stałego w celu opracowania nowych technik optymalizacji kształtu i topologii. Wprowadzono opis wzrostu jaki obserwujemy przy powstawaniu jednocześnie sztywnych i lekkich struktur biologicznych, z wykorzystaniem technik zaczerpniętych z teorii optymalizacji kształtu i topologii. Rozważono struktury szkieletowe z twardego materiału, stanowiące konstrukcję nośną materiału słabego. Proces wzrostu dotyczy szkieletu i polega na pojawianiu się w słabym materiale obszarów zbudowanych z materiału twardego oraz na powiększaniu się obszaru zajmowanego przez ten materiał. Tą ewolucję opisujemy wzdłuż linii czasu. Zatem podobszar wypełniony słabym materiał jest źródłem składnika potrzebnego do wzrostu szkieletu. Zakładamy, że obszary wypełnione twardym materiałem pojawiające się w obszarze słabym maja formę kul o małych promieniach i lokalizacja ich zarodkowania wynika z warunku minimalizacji zastępczej sztywności szkieletu. Położenie nowego obszaru zbrojenia materiałem twardym jest wyznaczane przez rozwiązanie zadania minimalizacji pochodnej topologicznej zastępczej sztywności szkieletu. W ten sposób otrzymuje się model wzrostu szkieletu z możliwością generacji nowych składników szkieletu poprzez optymalizację jego topologii jako podobszaru fazy słabej. Wzrost szkieletu przy zadanej topologii odbywa się zgodnie z prawem wzrostu podlegającym zasadzie zachowania masy ze źródłem o stałej wydajności. Otrzymany model matematyczny procesu wzrostu wykorzystuje technikę optymalizacji kształtu i topologii do wyznaczenia kształtu struktury poprzez minimalizację jej sztywności przy zmianie topologii szkieletu. W celu zastąpienia osobliwych zaburzeń obszaru całkowania przy zarodkowaniu nowych składników szkieletu przez zaburzenia nieosobliwe, w optymalizacji topologii ciała można wiec stosować znaną technikę dekompozycji obszaru całkowania i operator Steklova-Poincare. Z matematycznego punktu widzenia praca dotyczy modelu quazistatycznego ciała sprężystego o zmiennej geometrii. Zmiany geometrii podlegają optymalizacji kształtu i topologii ze względu na funkcjonał jakości jakim jest sztywność konstrukcji w każdym kroku czasowym rozpatrywanego okresu wzrostu.

12

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

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2009

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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.

13

Optimal internal dissipation of a damped wave equation using a topological approach

Münch A

International Journal of Applied Mathematics and Computer Science

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2009

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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.

14

Electrical impedance tomography: from topology to shape

Hintermuller M., Laurain A.

Control and Cybernetics

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2008

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Vol. 37, no 4

913-933

EN

A level set based shape and topology optimization approach to electrical impedance tomography (EIT) problems with piecewise constant conductivities is introduced. The proposed solution algorithm is initialized by using topological sensitivity analysis. Then it relies on the notion of shape derivatives to update the shape of the domains where conductivity takes different values.

15

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

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2007

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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.

16

Application of finite variations to topology and shape optimization of 2D structures

Bojczuk D., Sztebelak W.

Journal of Theoretical and Applied Mechanics

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2006

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Vol. 44 nr 2

323-349

EN

The method of simultaneous topology and shape optimization of 2D structures by finite topology modification is presented in the paper. Both, structures in a plane state of stress and bending Kirchhoff's plates are analyzed here. Conditions for the introduction of finite topology modification based on the topological derivative are specified. When the respective condition is satisfied, finite holes and finite variations of existing boundaries are introduced into the structure. Next, standard shape optimization of new holes and variable boundaries is performed. Two basic types of modification are considered here, namely the introduction of holes of a prescribed size and shape and the introduction of holes of an unknown size and shape together with the introduction of finite changes of other boundaries. A heuristic algorithm for optimal design of topology and shape is proposed in the paper. Illustrative examples confirm applicability of the proposed approach.

PL

W pracy rozpatrywana jest metoda jednoczesnej optymalizacji topologii i kształtu konstrukcji dwuwymiarowych przy użyciu skończonych modyfikacji topologii. Rozważania dotyczą zarówno konstrukcji tarczowych pracujących w płaskim stanie naprężenia, jak i płyt Kirchhoffa pracujących w stanie zgięciowym. Przy wykorzystaniu pochodnej topologicznej wyprowadzono warunki wprowadzania skończonych modyfikacji topologii. Gdy spełniony jest odpowiedni warunek modyfikacji, do konstrukcji wprowadzane są otwory o skończonych wymiarach oraz ewentualnie skończone modyfikacje pozostałych brzegów. Następnie wykonywana jest standardowa optymalizacja kształtu otworów i brzegów zewnętrznych. Analizowane są dwa podstawowe typy modyfikacji, a mianowicie wprowadzanie otworów o zadanej wielkości i kształcie oraz wprowadzanie otworów o nieznanej wielkości i kształcie wraz z ewentualną skończoną zmianą pozostałych brzegów. W pracy sformułowano odpowiedni algorytm heurystyczny optymalizacji topologii i kształtu rozpatrywanych konstrukcji. Przedstawione przykłady ilustracyjne potwierdzają przydatność zaproponowanego podejścia.

17

Topological derivative for linear elastic plate bending problems

Novotny A. A., Feijoo R. A., Padra C., Taroco E.

Control and Cybernetics

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2005

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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.

18

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

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2005

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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.

19

Topological derivative for optimal control problems

Sokołowski J., Żochowski A.

Control and Cybernetics

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1999

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Vol. 28, no 3

611-625

EN

The topological derivative is introduced for the extremal values of cost functionals for control problems. The optimal control problem considered in the paper is defined for the elliptic equation which models the deflection of an elastic membrane. The derivative measures the sensitivity of the optimal value of the cost with respect to changes in topology. A change in topology means removing a small ball from the interior of the domain of integration. The topological derivative can be used for obtaining the numerical solutions of the shape optimization problems.