Flip procedure in geometric approximation of multiple-component shapes – Application to multiple-inclusion detection

Bonnelie, Pierre; Bourdin, Loïc; Caubet, Fabien; Ruatta, Olivier

The SMAI Journal of computational mathematics, Tome 2 (2016), pp. 255-276.

We are interested in geometric approximation by parameterization of two-dimensional multiple-component shapes, in particular when the number of components is a priori unknown. Starting a standard method based on successive shape deformations with a one-component initial shape in order to approximate a multiple-component target shape usually leads the deformation flow to make the boundary evolve until it surrounds all the components of the target shape. This classical phenomenon tends to create double points on the boundary of the approximated shape.

In order to improve the approximation of multiple-component shapes (without any knowledge on the number of components in advance), we use in this paper a piecewise Bézier parameterization and we consider two procedures called intersecting control polygons detection and flip procedure. The first one allows to prevent potential collisions between two parts of the boundary of the approximated shape, and the second one permits to change its topology by dividing a one-component shape into a two-component shape.

For an experimental purpose, we include these two processes in a basic geometrical shape optimization algorithm and test it on the classical inverse obstacle problem. This new approach allows to obtain a numerical approximation of the unknown inclusion, detecting both the topology (i.e. the number of connected components) and the shape of the obstacle. Several numerical simulations are performed.

Classification : 68U05, 68W25, 49Q10, 65N21
Mots clés : Shape approximation; free-form shapes; multiple-component shapes; Bézier curves; intersecting control polygons detection; flip procedure; inverse obstacle problem; shape optimization

[1] Afraites, L.; Dambrine, M.; Eppler, K.; Kateb, D. Detecting perfectly insulated obstacles by shape optimization techniques of order two, Discrete Contin. Dyn. Syst. Ser. B, Volume 8 (2007) no. 2, p. 389-416 (electronic) | DOI | MR | Zbl

[2] Allaire, G.; Dapogny, C.; Frey, P. Shape optimization with a level set based mesh evolution method, Comput. Methods Appl. Mech. Engrg., Volume 282 (2014), pp. 22-53 https://dx-doi-org.docadis.ups-tlse.fr/10.1016/j.cma.2014.08.028 | DOI | MR | Zbl

[3] Allaire, G.; de Gournay, F.; Jouve, F.; Toader, A.-M. Structural optimization using topological and shape sensitivity via a level set method, Control Cybernet., Volume 34 (2005) no. 1, pp. 59-80 | MR | Zbl

[4] Ammari, H.; Kang, H. Reconstruction of small inhomogeneities from boundary measurements, Lecture Notes in Mathematics, 1846, Springer-Verlag, Berlin, 2004, x+238 pages https://dx-doi-org.docadis.ups-tlse.fr/10.1007/b98245 | DOI | MR | Zbl

[5] Badra, M.; Caubet, F.; Dambrine, M. Detecting an obstacle immersed in a fluid by shape optimization methods, Math. Models Methods Appl. Sci., Volume 21 (2011) no. 10, pp. 2069-2101 | DOI | MR | Zbl

[6] Bishop, E. A generalization of the Stone-Weierstrass theorem., Pacific J. Math., Volume 11 (1961) no. 3, pp. 777-783 http://projecteuclid.org/euclid.pjm/1103037116 | DOI | MR | Zbl

[7] Bourgeois, L.; Dardé, J. A quasi-reversibility approach to solve the inverse obstacle problem, Inverse Probl. Imaging, Volume 4 (2010) no. 3, pp. 351-377 | DOI | MR

[8] Burger, M.; Hackl, B.; Ring, W. Incorporating topological derivatives into level set methods, J. Comput. Phys., Volume 194 (2004) no. 1, pp. 344-362 | DOI | MR | Zbl

[9] Burger, M.; Osher, S. J. A survey on level set methods for inverse problems and optimal design, European J. Appl. Math., Volume 16 (2005) no. 2, pp. 263-301 https://dx-doi-org.docadis.ups-tlse.fr/10.1017/S0956792505006182 | DOI | MR | Zbl

[10] Caubet, F. Instability of an inverse problem for the stationary Navier-Stokes equations, SIAM J. Control Optim., Volume 51 (2013) no. 4, pp. 2949-2975 https://dx-doi-org.docadis.ups-tlse.fr/10.1137/110836857 | DOI | MR | Zbl

[11] Caubet, F.; Conca, C.; Godoy, M. On the detection of several obstacles in 2D Stokes flow: topological sensitivity and combination with shape derivatives, Inverse Probl. Imaging, Volume 10 (2016) no. 2, pp. 327-367 | DOI | MR | Zbl

[12] Caubet, F.; Dambrine, M. Localization of small obstacles in Stokes flow, Inverse Problems, Volume 28 (2012) no. 10, 105007 pages http://stacks.iop.org/0266-5611/28/i=10/a=105007 | DOI | MR | Zbl

[13] Caubet, F.; Dambrine, M.; Kateb, D. Shape optimization methods for the inverse obstacle problem with generalized impedance boundary conditions, Inverse Problems, Volume 29 (2013) no. 11, 115011, 26 pages | DOI | MR | Zbl

[14] Caubet, F.; Dambrine, M.; Kateb, D.; Timimoun, C. Z. A Kohn-Vogelius formulation to detect an obstacle immersed in a fluid, Inverse Probl. Imaging, Volume 7 (2013) no. 1, pp. 123-157 https://dx-doi-org.docadis.ups-tlse.fr/10.3934/ipi.2013.7.123 | DOI | MR | Zbl

[15] Christiansen, A. N.; Nobel-Jørgensen, M.; Aage, N.; Sigmund, O.; Bærentzen, J. A. Topology optimization using an explicit interface representation, Structural and Multidisciplinary Optimization, Volume 49 (2014) no. 3, pp. 387-399 | DOI | MR

[16] Colton, D.; Kress, R. Inverse acoustic and electromagnetic scattering theory, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1998, xii+334 pages | DOI | MR | Zbl

[17] Dardé, J. Quasi-reversibility and level set methods applied to elliptic inverse problems, Université Paris-Diderot - Paris VII, December (2010) https://pastel.archives-ouvertes.fr/tel-00551853 (Theses)

[18] Ericson, C. Real-Time Collision Detection (The Morgan Kaufmann Series in Interactive 3-D Technology) (The Morgan Kaufmann Series in Interactive 3D Technology), Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 2004

[19] Farin, G. Curves and Surfaces for CAGD: A Practical Guide, Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 2002

[20] Frey, P.J.; George, P.L. Mesh Generation: Application to Finite Elements, Second Edition, Wiley-ISTE, 2008

[21] Garreau, S.; Guillaume, P.; Masmoudi, M. The topological asymptotic for PDE systems: the elasticity case, SIAM J. Control Optim., Volume 39 (2001) no. 6, p. 1756-1778 (electronic) https://dx-doi-org.docadis.ups-tlse.fr/10.1137/S0363012900369538 | DOI | MR | Zbl

[22] Hecht, F. Finite Element Library Freefem++., http://www.freefem.org/ff++/

[23] Henrot, A.; Pierre, M. Variation et optimisation de formes : Une analyse géométrique., Mathématiques & Applications (Berlin) [Mathematics & Applications], 48, Springer, Berlin, 2005, xii+334 pages | Zbl

[24] Isakov, V. Inverse problems for partial differential equations, 127, Springer Science & Business Media, 2006 | MR | Zbl

[25] Kichenassamy, S.; Kumar, A.; Olver, P.; Tannenbaum, A.; Yezzi Jr., A. Gradient Flows and Geometric Active Contour Models, 1994

[26] Labbani-I., O.; Merveilleux-O., P.; Ruatta, O. Free Form based active contours for image segmentation and free space perception (preprint)

[27] Marco, A.; Martínez, J.-J. A fast and accurate algorithm for solving Bernstein–Vandermonde linear systems, Linear Algebra and its Applications, Volume 422 (2007) no. 2, pp. 616 -628 http://www.sciencedirect.com/science/article/pii/S0024379506005210 | DOI | MR | Zbl

[28] O’Rourke, J. Computational Geometry in C, Cambridge University Press, New York, NY, USA, 1998 | Zbl

[29] Pantz, O.; Trabelsi, K. Simultaneous shape, topology, and homogenized properties optimization, Struct. Multidiscip. Optim., Volume 34 (2007) no. 4, pp. 361-365 | DOI | MR | Zbl

[30] Persson, P.-O. Mesh Generation for Implicit Geometries, Massachusetts Institute of Technology (USA), Cambridge, MA, USA (2005) (Ph. D. Thesis AAI0807802) | MR

[31] Pjan, T.D.M. 3D Free Form Method and Applications to Robotics, University of Limoges (2014) (Masters thesis http://www.unilim.fr/pages_perso/olivier.ruatta/pub/stage.pdf)

[32] Santosa, F. A level-set approach for inverse problems involving obstacles., ESAIM, Control Optim. Calc. Var., Volume 1 (1996), pp. 17-33 | DOI | Numdam | Zbl

[33] Schumacher, A. Topologieoptimisierung von Bauteilstrukturen unter Verwendung von Lopchpositionierungkrieterien, Universität-Gesamthochschule-Siegen (1995) (Ph. D. Thesis)

[34] Sederberg, T. W. Computer aided geometric design, 2014

[35] Sethian, J. A. Level set methods and fast marching methods. Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science., Cambridge: Cambridge University Press, 1999, xx + 378 pages | Zbl

[36] Sokołowski, J.; Żochowski, A. On the topological derivative in shape optimization, SIAM J. Control Optim., Volume 37 (1999) no. 4, p. 1251-1272 (electronic) | DOI | MR | Zbl

[37] Sokołowski, J.; Zolésio, J.-P. Introduction to shape optimization, Springer Series in Computational Mathematics, 16, Springer-Verlag, Berlin, 1992, ii+250 pages (Shape sensitivity analysis) | MR