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Zastosowanie metody pochodnej topologicznej w elektrycznej tomografii impedancyjnej
Języki publikacji
Abstrakty
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 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.
Rocznik
Tom
Strony
4--8
Opis fizyczny
Bibliogr. 60 poz., rys.
Twórcy
autor
- Laboratório Nacional de Computação Científica LNCC/MCT, Coordenação de Matemática Aplicada e Computacional
autor
- Laboratório Nacional de Computação Científica LNCC/MCT, Coordenação de Matemática Aplicada e Computacional
autor
- Université de Lorraine, CNRS, INRIA, Institute Élie Cartan Nancy
Bibliografia
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- [15] Bojczuk D., Mróz Z.: Topological sensitivity derivative and finite topology modications: application to optimization of plates in bending. Structural and Multidisciplinary Optimization 39, 2009, 1–15.
- [16] Bonnet M.: Higher-order topological sensitivity for 2-D potential problems. International Journal of Solids and Structures 46(11–12), 2009, 2275–2292.
- [17] Burger M., Hackl B., Ring W.: Incorporating topological derivatives into level set methods. Journal of Computational Physics 194(1), 2004, 344–362.
- [18] Calderón A.P.: On an inverse boundary value problem. Computational and Applied Mathematics 25(2–3), 2006. Reprinted from the Seminar on Numerical Analysis and its Applications to Continuum Physics, Sociedade Brasileira de Matemática, Rio de Janeiro, 1980.
- [19] Canelas A., Laurain A., Novotny A.A.: A new reconstruction method for the inverse potential problem. Journal of Computational Physics 268, 2014, 417–431.
- [20] Canelas A., Laurain A., Novotny A.A.: A new reconstruction method for the inverse source problem from partial boundary measurements. Inverse Problems 31(7), 2015, 075009.
- [21] Canelas A., Novotny A.A., Roche J.R.: A new method for inverse electromagnetic casting problems based on the topological derivative. Journal of Computational Physics 230, 2011, 3570–3588.
- [22] de Faria J.R., Novotny A.A.: On the second order topological asymptotic expansion. Structural and Multidisciplinary Optimization 39(6), 2009, 547–555.
- [23] Feijóo G.R.: A new method in inverse scattering based on the topological derivative. Inverse Problems 20(6), 2004, 1819–1840.
- [24] Feijáo R.A., Novotny A.A., Taroco E., Padra C.: The topological derivative for the Poisson's problem. Mathematical Models and Methods in Applied Sciences 13(12), 2003, 1825–1844.
- [25] Garreau S., Guillaume Ph., Masmoudi M.: The topological asymptotic for PDE systems: the elasticity case. SIAM Journal on Control and Optimization 39(6), 2001, 1756–1778.
- [26] Giusti S.M., Novotny A.A., de Souza Neto E.A.: Sensitivity of the macroscopic response of elastic microstructures to the insertion of inclusions. Proceeding of the Royal Society A: Mathematical, Physical and Engineering Sciences 466, 2010, 1703–1723.
- [27] Giusti S.M., Novotny A.A., de Souza Neto E.A., Feijóo R.A.: Sensitivity of the macroscopic elasticity tensor to topological microstructural changes. Journal of the Mechanics and Physics of Solids 57(3), 2009, 555–570.
- [28] Giusti S.M., Novotny A.A., de Souza Neto E.A., Feijóo R.A.: Sensitivity of the macroscopic thermal conductivity tensor to topological microstructural changes. Computer Methods in Applied Mechanics and Engineering 198(5–8), This copy is for personal use only - distribution prohibited. 2009, 727–739, [DOI: 10.1016/j.cma.2008.10.005].
- [29] Giusti S.M., Novotny A.A., Sokołowski J.: Topological derivative for steadystate orthotropic heat diffusion problem. Structural and Multidisciplinary Optimization 40(1), 2010, 53–64.
- [30] Guzina B.B., Bonnet M.: Small-inclusion asymptotic of misfit functionals for inverse problems in acoustics. Inverse Problems 22(5), 2006, 1761–1785.
- [31] Hintermüller M.: Fast level set based algorithms using shape and topological sensitivity. Control and Cybernetics 34(1), 2005, 305–324.
- [32] Hintermüller M., Laurain A.: Electrical impedance tomography: from topology to shape. Control and Cybernetics 37(4), 2008, 913–933.
- [33] Hintermüller M., Laurain A.: Multiphase image segmentation and modulation recovery based on shape and topological sensitivity. Journal of Mathematical Imaging and Vision 35, 2009, 1–22.
- [34] Hintermüller M., Laurain A., Novotny A.A.: Second-order topological expansion for electrical impedance tomography. Advances in Computational Mathematics 36(2), 2012, 235–265.
- [35] Hlaváček I., Novotny A.A., Sokołowski J., Żochowski A.: On topological derivatives for elastic solids with uncertain input data. Journal of Optimization Theory and Applications 141(3), 2009, 569–595.
- [36] Jackowska-Strumiłło L., Sokołowski J., Żochowski A., Henrot A.: On numerical solution of shape inverse problems. Computational Optimization and Applications 23(2), 2002, 231–255.
- [37] Khludnev A.M., Novotny A.A., Sokołowski J., Żochowski A.: Shape and topology sensitivity analysis for cracks in elastic bodies on boundaries of rigid inclusions. Journal of the Mechanics and Physics of Solids 57(10), 2009, 1718–1732.
- [38] Kobelev V.: Bubble-and-grain method and criteria for optimal positioning inhomogeneities in topological optimization. Structural and Multidisciplinary Optimization 40(1–6), 2010, 117–135.
- [39] Larrabide I., Feijóo R.A., Novotny A.A., Taroco E.: Topological derivative: a tool for image processing. Computers & Structures 86(13–14), 2008, 1386–1403.
- [40] Leugering G., Sokołowski J.: Topological derivatives for elliptic problems on graphs. Control and Cybernetics 37, 2008, 971–998.
- [41] Lewinski T., Sokołowski J.: Energy change due to the appearance of cavities in elastic solids. International Journal of Solids and Structures 40(7), 2003, 1765–1803.
- [42] Masmoudi M., Pommier J., Samet B.: The topological asymptotic expansion for the Maxwell equations and some applications. Inverse Problems 21(2), 2005, 547–564.
- [43] Nazarov S.A., Sokołowski J.: Asymptotic analysis of shape functionals. Journal de Mathématiques Pures et Appliquées 82(2), 2003, 125–196.
- [44] Nazarov S.A., Sokołowski J.: Self-adjoint extensions of differential operators in application to shape optimization. Comptes Rendus Mecanique 331, 2003, 667–672.
- [45] Nazarov S.A., Sokołowski J.: Singular perturbations in shape optimization for the Dirichlet Laplacian. Comptes rendus - Mécanique 333(4), 2005, 305–310.
- [46] Nazarov S.A., Sokołowski J.: Self-adjoint extensions for the Neumann laplacian and applications. Acta Mathematica Sinica (English Series) 22(3), 2006, 879–906.
- [47] Nazarov S.A., Sokołowski J.: On asymptotic analysis of spectral problems in elasticity. Latin American Journal of Solids and Structures 8, 2011, 27–54.
- [48] Novotny A.A., Feijóo R.A., Padra C., Taroco E.: Topological sensitivity analysis. Computer Methods in Applied Mechanics and Engineering 192(7–8), 2003, 803–829.
- [49] Novotny A.A., Feijóo R.A., Padra C., Taroco E.: Topological derivative for linear elastic plate bending problems. Control and Cybernetics 34(1), 2005, 339–361.
- [50] Novotny A.A., Feijóo R.A., Taroco E., Padra C.: Topological sensitivity analysis for three dimensional linear elasticity problem. Computer Methods in Applied Mechanics and Engineering 196(41–44), 2007, 4354–4364.
- [51] Novotny A.A., Sokołowski J.: Topological derivatives in shape optimization. Interaction of Mechanics and Mathematics. Springer-Verlag, Berlin, Heidelberg 2013.
- [52] Novotny A.A., Sokołowski J., de Souza Neto E.A.: Topological sensitivity analysis of a multiscale constitutive model considering a cracked microstructure. Mathematical Methods in the Applied Sciences 33(5), 2010, 676–686.
- [53] Sankowski D., Sikora J.: Electrical Capacitance Tomography: Theoretical Basis and Applications. Electrotechnical Institute, Poland, Miedzylesie 2010.
- [54] Sikora J.: Boundary Element Method for Impedance and Optical Tomography. Oficyna Wydawnicza Politechniki Warszawskiej, Poland, Warsaw 2007.
- [55] Sikora J., Wójtowicz S.: Industrial and Biological Tomography: Theoretical Basis and Applications. Electrotechnical Institute, Poland, Miedzylesie 2010.
- [56] Sokołowski J., Żochowski A.: On the topological derivative in shape optimization. SIAM Journal on Control and Optimization 37(4), 1999, 1251–1272.
- [57] Sokołowski J., Żochowski A.: Optimality conditions for simultaneous topology and shape optimization. SIAM Journal on Control and Optimization 42(4), 2003, 1198–1221.
- [58] Sokołowski J., Żochowski A.: Modelling of topological derivatives for contact problems. Numerische Mathematik 102(1), 2005, 145–179.
- [59] Turevsky I., Gopalakrishnan S.H., Suresh K.: An ecient numerical method for computing the topological sensitivity of arbitrary-shaped features in plate bending. International Journal for Numerical Methods in Engineering 79(13), 2009, 1683–1702.
- [60] Van Goethem N., Novotny A.A.: Crack nucleation sensitivity analysis. Mathematical Methods in the Applied Sciences 33(16), 2010, 1978–1994.
Uwagi
Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-4e3138ad-9bae-4b37-bd02-b30ce7d19472