Topological optimization of non-Newtonian fluids is currently an active area in applied mathematics. Topological optimization aims to give an asymptotic expansion of a shape functional under PDE constraints. In this paper, we propose a topological derivative formulation for a non-Newtonian fluid governed by nonlinear PDE. By using generalized adjoint method, we give an asymptotic expansion of a given functional and consequently some numerical simulations are given in domains with obstacles and domains without obstacles.

non-Newtonian fluids, topological optimisation, asymptotic expansion, numerical method, numerical simulations.

Received: September 7, 2022; Accepted: October 17, 2022; Published: October 27, 2022

How to cite this article: Alassane Sy, Serigne Diouf and Mouhamadou Ngom, Topological optimization for a quasi-linear problem with application to non-Newtonian fluids, International Journal of Numerical Methods and Applications 22 (2022), 67-85. http://dx.doi.org/10.17654/0975045222007

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] D. Auroux, M. Masmoudi and L. Belaid, Image restoration and classification by topological asymptotic expansion, Variational Formulations in Mechanics: Theory and Applications, E. Taroco, E. A. de Souza Neto and A. A. Novotny, eds., CIMNE, Barcelona, Spain, 2006.
[2] S. Amstutz, Aspects théoriques et numériques en optimisation de forme topologique, Thèse d’Etat, INSA Toulouse, 2003.
[3] S. Amstutz, Topological sensitivity analysis for some nonlinear PDE system, J. Math. Pures Appl. 85 (2006), 540-570.
[4] D. Auroux, From restoration by topological gradient to medical image segmentation via an asymptotic expansion, Math. Comput. Model. 49(11-12) (2009), 2191-2205.
[5] R. P. Agarwal, M. E. Filippakis, D. O’Regan and N. S. Papageorgiou, Constant sign and nodal solutions for problems with the p-Laplacian and a nonsmooth potential using variational techniques, Bound. Value Probl. (2009), 32 pages, Article ID 820237.
[6] J. Céa, Conception optimale ou identification de formes, calcul rapide de la dérivée directionnelle de la fonction coût, RAIRO, Modél. Math. Anal. Numér. 20 (1986), 371-402.
[7] S. Diouf, M. M. Diop and A. Sy, Image processing using nonlinear optimization, International Journal of Numerical Methods and Applications 21(1) (2021), 1-16. http://dx.doi.org/10.17654/NM020010001.
[8] S. Oruganti and R. Shivaji, Existence results for classes of P-Laplacian semiposition equations, Bound. Value Probl. (2006), 7 pages, Article ID 87483.
[9] A. Sy and D. Seck, Topological optimisation for the p-Laplacian operator and an application in image processing, Bound. Value Probl. (2009), 20 pages, Article ID 986813.
[10] N. Ju, Numerical analysis of parabolic p-Laplacian: approximation of trajectories, SIAM J. Numer. Anal. 37(6) (2000), 1861-1884.
[11] R. I. Tanner, Engineering Rheology, Oxford University Press, New York, 1985.
[12] C. Rigal, Comportement de fluides complexes sous écoulement: approche exprimentale par résonance magnétique nuclaire et techniques optiques et simulations numériques, Thèse, Institut National Polytechnique de Lorraine, 2012.
[13] A. Gueye, Modélisation et simulations numériques des écoulements et instabilités thermiques de fluides non Newtoniens en milieu poreux, Thèse, Universit Lille 1, 2015.
[14] G. Ella Eny, Instabilités thermiques et thermodiffusives de fluides viscoelastiques saturant en milieux poureux, Thèse, Universit Lille 1, Sciences et Technologie, 2011.
[15] J. Bear and M. Y. Corapcioglu, Advances in transport phenomena in porous media, Proceedings of the NATO Advanced Study Institute on “Fundamentals of Transport Phenomena in Porous Media”, Newark, Delaware, USA, July 14-23, 1985.
[16] A. Seck, A. Sy and D. Seck, Asymptotic analysis of a shape functional in incompressible Navier-Stokes flows, International Journal of Mathematical Analysis 13(7) (2019), 329-347. doi.org/10.12988/ijma.2019.9631.