# American Institute of Mathematical Sciences

November  2018, 17(6): 2683-2702. doi: 10.3934/cpaa.2018127

## Shape optimization approach for solving the Bernoulli problem by tracking the Neumann data: A Lagrangian formulation

 Department of Mathematics and Computer Science, College of Science, University of the Philippines Baguio, Gov. Pack Rd., Baguio City 2600, Philippines

* Corresponding author

Received  March 2017 Revised  August 2017 Published  June 2018

Fund Project: This work was an output from a project funded by the UP System Emerging Interdisciplinary Research (EIDR) Program (OVPAA-EIDR-C05-015)

The exterior Bernoulli free boundary problem is considered and reformulated into a shape optimization setting wherein the Neumann data is being tracked. The shape differentiability of the cost functional associated with the formulation is studied, and the expression for its shape derivative is established through a Lagrangian formulation coupled with the velocity method. Also, it is illustrated how the computed shape derivative can be combined with the modified $H^1$ gradient method to obtain an efficient algorithm for the numerical solution of the shape optimization problem.

Citation: Julius Fergy T. Rabago, Jerico B. Bacani. Shape optimization approach for solving the Bernoulli problem by tracking the Neumann data: A Lagrangian formulation. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2683-2702. doi: 10.3934/cpaa.2018127
##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, London, 1975.Google Scholar [2] H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math., 325 (1981), 105-144. Google Scholar [3] H. Azegami, A solution to domain optimization problems, Trans of Japan Soc. of Mech. Engs., Ser. A, 60 (1994), 1479-1486 (in Japanese).Google Scholar [4] H. Azegami and Z. Q. Wu, Domain optimization analysis in linear elastic problems: approach using traction method, JSME Int J., Ser. A, 39 (1996), 272-278. Google Scholar [5] H. Azegami, S. Kaizu, M. Shimoda and E. Katamine, Irregularity of shape optimization problems and an improvement technique, in Computer Aided Optimization Design of Structures V (S. Hernandez and C. A. Brebbia eds.), Computational Mechanics Publications, Southampton, (1997), 309-326.Google Scholar [6] H. Azegami and Z. Takeuchi, A smoothing method for shape optimization: traction method using the Robin condition, Int. J. Comp. Meth-Sing., 3 (2006), 21-33. Google Scholar [7] H. Azegami, S. Fukumoto and T. Aoyama, Shape optimization of continua using nurbs as basis functions, Struct. Multidiscipl. Optimiz., 47 (2013), 247-258. Google Scholar [8] H. Azegami, L. Zhou, K. Umemura and N. Kondo, Shape optimization for a link mechanism, Struct. Multidiscipl. Optimiz., 48 (2013), 115-125. Google Scholar [9] B. Abda, F. Bouchon, G. Peichl, M. Sayeh and R. Touzani, A new formulation for the Bernoulli problem, in Proceedings of the 5th International Conference on Inverse Problems, Control and Shape Optimization, (2010), 1-19.Google Scholar [10] J. Bacani, Methods of Shape Optimization in Free Boundary Problems, Ph. D. Thesis, Karl-Franzens-Universität Graz, Graz, Austria, 2013.Google Scholar [11] J. B. Bacani and G. H. Peichl, On the first-order shape derivative of the Kohn-Vogelius cost functional of the Bernoulli problem, Abstr. Appl. Anal., 2013 (2013), Article ID 384320, 19 pp.Google Scholar [12] Z. Belhachmi and H. Meftahi, Shape sensitivity analysis for an interface problem via minimax differentiability, Appl.Math. Comput., 219 (2013), 6828-6842. Google Scholar [13] J. Céa, Numerical methods of shape optimal design, in Optimization of Distributed Parameter Structures 2 (E. J. Haug and J. Céa eds.), Sijthoff and Noordhoff, Alphen aan den Rijn, (1981), 1049-1088.Google Scholar [14] J. Céa, Conception optimale ou identification de formes, calcul rapide de la derivee dircetionelle de la fonction cout, Math. Mod. Numer. Anal., 20 (1986), 371-402. Google Scholar [15] D. Chenais, On the existence of a solution in a domain identification problem, J. Math. Anal. Appl., (1975), 189-219. Google Scholar [16] R. Correa and A. Seeger, Directional derivative of a minimax function, Nonlinear Anal., 9 (1985), 13-22. Google Scholar [17] M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, 2$^{nd}$ ed., Adv. Des. Control 22, SIAM, Philadelphia, 2011.Google Scholar [18] M. C. Delfour and J.-P. Zolésio, Shape sensitivity analysis via min max differentiability, SIAM J. Control Optim., 26 (1988), 834-862. Google Scholar [19] M. C. Delfour and J.-P. Zolésio, Velocity method and Lagrangian formulation for the computation of the shape Hessian, SIAM J. Control Optim., 29 (1991), 1414-1442. Google Scholar [20] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland Publishing Co., Amsterdam, 1976. Translated from the French, Studies in Mathematics and its Applications, Vol. 1.Google Scholar [21] K. Eppler and H. Harbrecht, On a Kohn-Vogelius like formulation of free boundary problems, Comput. Optim. Appl., 52 (2012), 69-85. Google Scholar [22] K. Eppler and H. Harbrecht, Tracking Neumann data for stationary free boundary problems, SIAM J. Control Optim., 48 (2009), 2901-2916. Google Scholar [23] K. Eppler and H. Harbrecht, Tracking the Dirichlet data in $L^2$ is an ill-posed problem, J. Optim. Theory Appl., 145 (2010), 17-35. Google Scholar [24] K. Eppler and H. Harbrecht, Shape optimization for free boundary problems-analysis and numerics, in Constrained Optimization and Optimal Control for Partial Differential Equations, 160 (2012), 277-288.Google Scholar [25] L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, USA, 1998.Google Scholar [26] M. Flucher and M. Rumpf, Bernoulli's free-boundary problem, qualitative theory and numerical approximation, J. Reine Angew. Math., 486 (2003), 165-204. Google Scholar [27] P. Grisvard, Elliptic Problems in Non-smooth Domains, Pitman Publishing, Marshfield, Massachussetts, USA, 1985.Google Scholar [28] A. Friedman, Free boundary problems in science and technology, Notices of the AMS, 47 (2000), 854-861. Google Scholar [29] Z. Gao and Y. Ma, Shape gradient of the dissipated energy functional in shape optimization for the viscous incompressible flow, Appl Numer Math., 58 (2008), 1720-1741. Google Scholar [30] Z. Gao, Y. Ma and H. W. Zhuang, Shape Hessian for generalized Oseen flow by differentiability of a minimax: a Lagrangian approach, Czech. Math. J., 57 (2007), 987-1011. Google Scholar [31] D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1983.Google Scholar [32] J. Hadamard, Mémoire sur le probleme d’analyse relatif a l’équilibre des plaques élastiques, in Mémoire des savants étrangers, 33, 1907, Œuvres de Jacques Hadamard, editions du C. N. R. S., Paris, (1968), 515-641.Google Scholar [33] J. Haslinger, K. Ito, T. Kozubek, K. Kunisch and G. Peichl, On the shape derivative for problems of Bernoulli type, Interfaces Free Bound., 1 (2009), 317-330. Google Scholar [34] J. Haslinger, T. Kozubek, K. Kunisch and G. Peichl, Shape optimization and fictitious domain approach for solving free boundary problems of Bernoulli type, Comput. Optim. Appl., 26 (2003), 231-251. Google Scholar [35] J. Haslinger, T. Kozubek, K. Kunisch and G. Peichl, Fictitious domain methods in shape optimization with applications in free-boundary problems, Comput. Optim. Appl., 26 (2003), 231-251. Google Scholar [36] J. Haslinger, T. Kozubek, K. Kunisch and G. Peichl, An embedding domain approach for a class of 2-d shape optimization problems: mathematical analysis, J. Math. Anal. Appl., 290 (2004), 665-685. Google Scholar [37] F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265. Google Scholar [38] A. Henrot and A. Shangholian, Convexity of free boundaries with Bernoulli type boundary condition, Nonlinear Anal., 28 (1997), 815-823. Google Scholar [39] M. H. Imam, Three dimensional shape optimization, Int. J. Num. Meth. Engrg., 18 (1982), 661-673. Google Scholar [40] H. Kasumba, Shape optimization approaches to free-surface problems, Int. J. Numer. Meth. Fluids, 74 (2014), 818-845. Google Scholar [41] K. Ito, K. Kunisch and G. Peichl, Variational approach to shape derivatives, ESAIM Control Optim. Calc. Var., 14 (2008), 517-539. Google Scholar [42] K. Ito, K. Kunisch and G. Peichl, Variational approach to shape derivative for a class of Bernoulli problem, J. Math. Anal. Appl., 314 (2006), 126-149. Google Scholar [43] A. Laurain and H. Meftahi, Shape and parameter reconstruction for the Robin inverse problem, J. Inverse Ill-Posed Probl., 24 (2016), 643-662. Google Scholar [44] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Berlin Heidelberg: Springer-Verlag, 1971.Google Scholar [45] H. Meftahi, Stability analysis in the inverse Robin transmission problem, Math. Methods Appl. Sci., 40 (2016), 2505-2521. Google Scholar [46] J. Neuberger, in Sobolev Gradients and Differential Equations (J-M. Morel and B. Teissier eds.), Lecture Notes in Mathematics. Springer: Berlin, 2010.Google Scholar [47] H. Meftahi and J.-P. Zolésio, Sensitivity analysis for some inverse problems in linear elasticity via minimax differentiability, Appl. Math. Model, 39 (2015), 1554-1576. Google Scholar [48] O. Pironneau and B. Mohammadi, Applied Shape Optimization in Fluid, Oxford University Press Inc: New York, 2001.Google Scholar [49] J. F. T. Rabago, Shape Optimization for the Bernoulli Free Boundary Problem Via Céa's Classical Lagrange Method and Min-Max Differentiability of the Lagrangian, M. Sc. Thesis, University of the Philippines Baguio, Philippines, 2016.Google Scholar [50] J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization, Springer, Berlin, Germany, 1991.Google Scholar

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##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, London, 1975.Google Scholar [2] H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math., 325 (1981), 105-144. Google Scholar [3] H. Azegami, A solution to domain optimization problems, Trans of Japan Soc. of Mech. Engs., Ser. A, 60 (1994), 1479-1486 (in Japanese).Google Scholar [4] H. Azegami and Z. Q. Wu, Domain optimization analysis in linear elastic problems: approach using traction method, JSME Int J., Ser. A, 39 (1996), 272-278. Google Scholar [5] H. Azegami, S. Kaizu, M. Shimoda and E. Katamine, Irregularity of shape optimization problems and an improvement technique, in Computer Aided Optimization Design of Structures V (S. Hernandez and C. A. Brebbia eds.), Computational Mechanics Publications, Southampton, (1997), 309-326.Google Scholar [6] H. Azegami and Z. Takeuchi, A smoothing method for shape optimization: traction method using the Robin condition, Int. J. Comp. Meth-Sing., 3 (2006), 21-33. Google Scholar [7] H. Azegami, S. Fukumoto and T. Aoyama, Shape optimization of continua using nurbs as basis functions, Struct. Multidiscipl. Optimiz., 47 (2013), 247-258. Google Scholar [8] H. Azegami, L. Zhou, K. Umemura and N. Kondo, Shape optimization for a link mechanism, Struct. Multidiscipl. Optimiz., 48 (2013), 115-125. Google Scholar [9] B. Abda, F. Bouchon, G. Peichl, M. Sayeh and R. Touzani, A new formulation for the Bernoulli problem, in Proceedings of the 5th International Conference on Inverse Problems, Control and Shape Optimization, (2010), 1-19.Google Scholar [10] J. Bacani, Methods of Shape Optimization in Free Boundary Problems, Ph. D. Thesis, Karl-Franzens-Universität Graz, Graz, Austria, 2013.Google Scholar [11] J. B. Bacani and G. H. Peichl, On the first-order shape derivative of the Kohn-Vogelius cost functional of the Bernoulli problem, Abstr. Appl. Anal., 2013 (2013), Article ID 384320, 19 pp.Google Scholar [12] Z. Belhachmi and H. Meftahi, Shape sensitivity analysis for an interface problem via minimax differentiability, Appl.Math. Comput., 219 (2013), 6828-6842. Google Scholar [13] J. Céa, Numerical methods of shape optimal design, in Optimization of Distributed Parameter Structures 2 (E. J. Haug and J. Céa eds.), Sijthoff and Noordhoff, Alphen aan den Rijn, (1981), 1049-1088.Google Scholar [14] J. Céa, Conception optimale ou identification de formes, calcul rapide de la derivee dircetionelle de la fonction cout, Math. Mod. Numer. Anal., 20 (1986), 371-402. Google Scholar [15] D. Chenais, On the existence of a solution in a domain identification problem, J. Math. Anal. Appl., (1975), 189-219. Google Scholar [16] R. Correa and A. Seeger, Directional derivative of a minimax function, Nonlinear Anal., 9 (1985), 13-22. Google Scholar [17] M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, 2$^{nd}$ ed., Adv. Des. Control 22, SIAM, Philadelphia, 2011.Google Scholar [18] M. C. Delfour and J.-P. Zolésio, Shape sensitivity analysis via min max differentiability, SIAM J. Control Optim., 26 (1988), 834-862. Google Scholar [19] M. C. Delfour and J.-P. Zolésio, Velocity method and Lagrangian formulation for the computation of the shape Hessian, SIAM J. Control Optim., 29 (1991), 1414-1442. Google Scholar [20] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland Publishing Co., Amsterdam, 1976. Translated from the French, Studies in Mathematics and its Applications, Vol. 1.Google Scholar [21] K. Eppler and H. Harbrecht, On a Kohn-Vogelius like formulation of free boundary problems, Comput. Optim. Appl., 52 (2012), 69-85. Google Scholar [22] K. Eppler and H. Harbrecht, Tracking Neumann data for stationary free boundary problems, SIAM J. Control Optim., 48 (2009), 2901-2916. Google Scholar [23] K. Eppler and H. Harbrecht, Tracking the Dirichlet data in $L^2$ is an ill-posed problem, J. Optim. Theory Appl., 145 (2010), 17-35. Google Scholar [24] K. Eppler and H. Harbrecht, Shape optimization for free boundary problems-analysis and numerics, in Constrained Optimization and Optimal Control for Partial Differential Equations, 160 (2012), 277-288.Google Scholar [25] L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, USA, 1998.Google Scholar [26] M. Flucher and M. Rumpf, Bernoulli's free-boundary problem, qualitative theory and numerical approximation, J. Reine Angew. Math., 486 (2003), 165-204. Google Scholar [27] P. Grisvard, Elliptic Problems in Non-smooth Domains, Pitman Publishing, Marshfield, Massachussetts, USA, 1985.Google Scholar [28] A. Friedman, Free boundary problems in science and technology, Notices of the AMS, 47 (2000), 854-861. Google Scholar [29] Z. Gao and Y. Ma, Shape gradient of the dissipated energy functional in shape optimization for the viscous incompressible flow, Appl Numer Math., 58 (2008), 1720-1741. Google Scholar [30] Z. Gao, Y. Ma and H. W. Zhuang, Shape Hessian for generalized Oseen flow by differentiability of a minimax: a Lagrangian approach, Czech. Math. J., 57 (2007), 987-1011. Google Scholar [31] D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1983.Google Scholar [32] J. Hadamard, Mémoire sur le probleme d’analyse relatif a l’équilibre des plaques élastiques, in Mémoire des savants étrangers, 33, 1907, Œuvres de Jacques Hadamard, editions du C. N. R. S., Paris, (1968), 515-641.Google Scholar [33] J. Haslinger, K. Ito, T. Kozubek, K. Kunisch and G. Peichl, On the shape derivative for problems of Bernoulli type, Interfaces Free Bound., 1 (2009), 317-330. Google Scholar [34] J. Haslinger, T. Kozubek, K. Kunisch and G. Peichl, Shape optimization and fictitious domain approach for solving free boundary problems of Bernoulli type, Comput. Optim. Appl., 26 (2003), 231-251. Google Scholar [35] J. Haslinger, T. Kozubek, K. Kunisch and G. Peichl, Fictitious domain methods in shape optimization with applications in free-boundary problems, Comput. Optim. Appl., 26 (2003), 231-251. Google Scholar [36] J. Haslinger, T. Kozubek, K. Kunisch and G. Peichl, An embedding domain approach for a class of 2-d shape optimization problems: mathematical analysis, J. Math. Anal. Appl., 290 (2004), 665-685. Google Scholar [37] F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265. Google Scholar [38] A. Henrot and A. Shangholian, Convexity of free boundaries with Bernoulli type boundary condition, Nonlinear Anal., 28 (1997), 815-823. Google Scholar [39] M. H. Imam, Three dimensional shape optimization, Int. J. Num. Meth. Engrg., 18 (1982), 661-673. Google Scholar [40] H. Kasumba, Shape optimization approaches to free-surface problems, Int. J. Numer. Meth. Fluids, 74 (2014), 818-845. Google Scholar [41] K. Ito, K. Kunisch and G. Peichl, Variational approach to shape derivatives, ESAIM Control Optim. Calc. Var., 14 (2008), 517-539. Google Scholar [42] K. Ito, K. Kunisch and G. Peichl, Variational approach to shape derivative for a class of Bernoulli problem, J. Math. Anal. Appl., 314 (2006), 126-149. Google Scholar [43] A. Laurain and H. Meftahi, Shape and parameter reconstruction for the Robin inverse problem, J. Inverse Ill-Posed Probl., 24 (2016), 643-662. Google Scholar [44] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Berlin Heidelberg: Springer-Verlag, 1971.Google Scholar [45] H. Meftahi, Stability analysis in the inverse Robin transmission problem, Math. Methods Appl. Sci., 40 (2016), 2505-2521. Google Scholar [46] J. Neuberger, in Sobolev Gradients and Differential Equations (J-M. Morel and B. Teissier eds.), Lecture Notes in Mathematics. Springer: Berlin, 2010.Google Scholar [47] H. Meftahi and J.-P. Zolésio, Sensitivity analysis for some inverse problems in linear elasticity via minimax differentiability, Appl. Math. Model, 39 (2015), 1554-1576. Google Scholar [48] O. Pironneau and B. Mohammadi, Applied Shape Optimization in Fluid, Oxford University Press Inc: New York, 2001.Google Scholar [49] J. F. T. Rabago, Shape Optimization for the Bernoulli Free Boundary Problem Via Céa's Classical Lagrange Method and Min-Max Differentiability of the Lagrangian, M. Sc. Thesis, University of the Philippines Baguio, Philippines, 2016.Google Scholar [50] J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization, Springer, Berlin, Germany, 1991.Google Scholar
Initial and final shape of the annular domain $\Omega$
Initial and final shape of the annular domain $\Omega$
History of values of the cost functional $J$
Cost Values
 Iter. Cost(α = 0.001) Cost(α = 0.01) 1 128.187510 128.187510 2 0.19825995 0.36967179 3 0.06116257 0.00004442 4 0.00000512 0.00000004
 Iter. Cost(α = 0.001) Cost(α = 0.01) 1 128.187510 128.187510 2 0.19825995 0.36967179 3 0.06116257 0.00004442 4 0.00000512 0.00000004
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