ZASTOSOWANIE METODY POCHODNEJ TOPOLOGICZNEJ W ELEKTRYCZNEJ TOMOGRAFII IMPEDANCYJNEJ

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.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 kontekście 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

On topological derivative for contact problem in elasticity

In the paper the general method for shape-topology sensitivity analysis of contact problems is proposed. The method uses the domain decomposition method combined with the specific properties of minimizers for the energy functional. The method is applied to the static problem of an elastic body in frictionless contact with an rigid foundation. The contact model allows a finite interpenetration of the bodies on the contact region. This interpenetration is modeled by means of a scalar function that depends on the normal component of the displacement field on the potential contact zone. We present the asymptotic behavior of the energy shape functional when a spheroidal void is introduced in an arbitrary point of the elastic body. For the asymptotic analysis, we use the domain decomposition technique and the associated Steklov-Poincaré pseudodifferential operator. The differentiability of the energy with respect to the non-smooth perturbation is established. A closed form for the topological derivative is also presented

Asymptotic analysis and topological derivatives for shape and topology optimization of elasticity problems in two spactial dimensions

Topological derivatives for elasticity problems are used in shape and topology optimization in structural mechanics. We propose an approach to the asymptotic analysis of singular perturbations of geometrical domains. This approach can be used in order to determine the exact solutions of elasticity boundary value problems in domains with small holes, and determine the explicit asymptotic expansions of solutions with respect to small parameter which describes the radius of internal hole. The elastic potentials of Muskhelishvili gives us an explicite solution in the ring $C(\rho,R)=\{\rho < |x| < R \}$ in the form of complex valued series. The series depends on the small parameter, the radius $\rho$ of the ring, and we are interested in the behavior of the series for the passage $\rho\to 0$. Such analysis leads to the expansion of the elastic energy in the form $$\mathcal{E}(\rho,R)=\mathcal{E}(0,R)+\rho^2\mathcal{E}^1(R)+\rho^4\mathcal{E}^2(R)+\dots\ ,$$ where $\mathcal{E}^1(R)$ is used to determine the first order topological derivatives of shape functionals, and $\mathcal{E}^2(R)$ can be used to determine the second order topological derivatives of shape functionals. In the paper the first order term $\mathcal{E}^1(R)$ is given, however the method is general and can be used to determine the subsequent terms of the energy expansion and the topological derivatives of higher order

Shape and topology sensitivity analysis for cracks in elastic bodies on boundaries of rigid inclusions

We consider a 3D elastic body with a rigid inclusion and a crack located at the boundary of the inclusion. It is assumed that non-penetration conditions are imposed at the crack faces which do not allow the opposite crack faces to penetrate each other. We analyze the variational formulation of the problem and provide shape and topology sensitivity analysis of the solution

Topological derivatives for semilinear elliptic equations

International audienceThe form of topological derivatives for 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 $L^{\infty}$ norm are obtained. Results of numerical experiments which confirm the theoretical convergence rate are presented

Shape and topological sensitivity analysis in domains with cracks

Framework for shape and topology sensitivity analysis in geometrical domains with cracks is established for elastic bodies in two spatial dimensions. Equilibrium problem for elastic body with cracks is considered. Inequality type boundary conditions are prescribed at the crack faces providing a non-penetration between the crack faces. Modelling of such problems in two spatial dimensions is presented with all necessary details for further applications in shape optimization in structural mechanics. In the paper, general results on the shape and topology sensitivity analysis of this problem are provided. The results are interesting on its own. In particular, the existence of the shape and topological derivatives of the energy functional is obtained. It is shown, in fact, that the level set type method \cite{Fulman} can be applied to shape and topology opimization of the related variational inequalities for elasticity problems in domains with cracks, with the nonpenetration condition prescribed on the crack faces. The results presented in the paper can be used for numerical solution of shape optimization and inverse problems in structural mechanics

Applications of Asymptotic Analysis

This workshop focused on asymptotic analysis and its fundamental role in the derivation and understanding of the nonlinear structure of mathematical models in various fields of applications, its impact on the development of new numerical methods and on other fields of applied mathematics such as shape optimization. This was complemented by a review as well as the presentation of some of the latest developments of singular perturbation methods

Modelling of topological derivatives for contact problems

The problem of topology optimisation is considered for free boundary problems of thin obstacle types. The formulae for the first term of asymptotics for energy functionals are derived. The precision of obtained terms is verified numerically. The topological differentiability of solutions to variational inequalities is established. In particular, the so-called {\it outer asymptotic expansion} for solutions of contact problems in elasticity with respect to singular perturbation of geometrical domain depending on small parameter are derived by an application of nonsmooth analysis. Such results lead to the {\it topological derivatives} of shape functionals for contact problems. The topological derivatives are used in numerical methods of simultaneous shape and topology optimisation