## Control of a mechanical aeration process via topological sensitivity analysis.(English)Zbl 1162.76019

Summary: The topological sensitivity analysis method gives the variation of a criterion with respect to the creation of a small hole in the domain. In this paper, we use this method to control the mechanical aeration process in eutrophic lakes. A simplified model based on incompressible Navier-Stokes equations is used, only considering the liquid phase, which is the dominant one. The injected air is taken into account through local boundary conditions for the velocity, on the injector holes. A 3D numerical simulation of the aeration effects is proposed using a mixed finite element method. In order to generate the best motion in the fluid for aeration purposes, the optimization of the injector location is considered. The main idea is to carry out topological sensitivity analysis with respect to the insertion of an injector. Finally, a topological optimization algorithm is proposed and some numerical results, showing the efficiency of our approach, are presented.

### MSC:

 76D55 Flow control and optimization for incompressible viscous fluids 76D07 Stokes and related (Oseen, etc.) flows 76T10 Liquid-gas two-phase flows, bubbly flows 76M10 Finite element methods applied to problems in fluid mechanics
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### References:

 [1] M. Abdelwahed, M. Amara, F. Dabaghi, Numerical analysis of a two phase flow, J. Sci Comput. (in press); M. Abdelwahed, M. Amara, F. Dabaghi, Numerical analysis of a two phase flow, J. Sci Comput. (in press) · Zbl 1125.76397 [2] Abdelwahed, M.; Dabaghi, F.; Ouazar, D., A virtual numerical simulator for aeration effects in lake eutrophication, Int. J. Comput. Fluid. Dynamics, 16, 2, 119-128 (2002) · Zbl 1143.76472 [3] Arnold, D.; Brezzi, F.; Fortin, M., A stable finite element for the Stokes equations, Calcolo, 21, 4, 337-344 (1984) · Zbl 0593.76039 [4] E. Clement, Dispertion de bulles et modifications du mouvement de la phase porteuse dans des écoulements tourbillonnaires, Ph.D. Thesis, Institut Nationale polytechnique de Toulouse, 1999; E. Clement, Dispertion de bulles et modifications du mouvement de la phase porteuse dans des écoulements tourbillonnaires, Ph.D. Thesis, Institut Nationale polytechnique de Toulouse, 1999 [5] D. Legende, Quelques aspects des forces hydrodynamiques et des transferts de chaleur sur une bulle sphérique, Ph.D. Thesis, INP Toulouse, France, 1996; D. Legende, Quelques aspects des forces hydrodynamiques et des transferts de chaleur sur une bulle sphérique, Ph.D. Thesis, INP Toulouse, France, 1996 [6] Guillaume, Ph.; Sid Idris, K., Topological sensitivity and shape optimization for the Stokes equations, SIAM J. Control Optim., 43, 1, 1-31 (2004) · Zbl 1093.49029 [7] Hassine, M.; Jan, S.; Masmoudi, M., From differential calculus to 0-1 topological optimization, SIAM J. Control Optim., 45, 6, 1965-1987 (2007) · Zbl 1139.49039 [8] Hassine, M.; Masmoudi, M., The topological asymptotic expansion for the quasi-Stokes problem, ESAIM, COCV J., 10, 4, 478-504 (2004) · Zbl 1072.49027 [9] M. Ishii, Thermo-fluid dynamic theory of a two-phase flow, Collection de la direction des études de recherche d’électricité de France, EYROLLES, 1975; M. Ishii, Thermo-fluid dynamic theory of a two-phase flow, Collection de la direction des études de recherche d’électricité de France, EYROLLES, 1975 · Zbl 0325.76135 [10] Sokolowski, J.; Zochowski, A., On the topological derivative in shape optimization, SIAM J. Control Optim., 37, 4, 1251-1272 (1999) · Zbl 0940.49026
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