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.

[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.

[3]

H. Azegami, A solution to domain optimization problems, Trans of Japan Soc. of Mech. Engs., Ser. A, 60 (1994), 1479-1486 (in Japanese).

[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.

[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.

[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.

[7]

H. AzegamiS. Fukumoto and T. Aoyama, Shape optimization of continua using nurbs as basis functions, Struct. Multidiscipl. Optimiz., 47 (2013), 247-258.

[8]

H. AzegamiL. ZhouK. Umemura and N. Kondo, Shape optimization for a link mechanism, Struct. Multidiscipl. Optimiz., 48 (2013), 115-125.

[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.

[10]

J. Bacani, Methods of Shape Optimization in Free Boundary Problems, Ph. D. Thesis, Karl-Franzens-Universität Graz, Graz, Austria, 2013.

[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.

[12]

Z. Belhachmi and H. Meftahi, Shape sensitivity analysis for an interface problem via minimax differentiability, Appl.Math. Comput., 219 (2013), 6828-6842.

[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.

[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.

[15]

D. Chenais, On the existence of a solution in a domain identification problem, J. Math. Anal. Appl., (1975), 189-219.

[16]

R. Correa and A. Seeger, Directional derivative of a minimax function, Nonlinear Anal., 9 (1985), 13-22.

[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.

[18]

M. C. Delfour and J.-P. Zolésio, Shape sensitivity analysis via min max differentiability, SIAM J. Control Optim., 26 (1988), 834-862.

[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.

[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.

[21]

K. Eppler and H. Harbrecht, On a Kohn-Vogelius like formulation of free boundary problems, Comput. Optim. Appl., 52 (2012), 69-85.

[22]

K. Eppler and H. Harbrecht, Tracking Neumann data for stationary free boundary problems, SIAM J. Control Optim., 48 (2009), 2901-2916.

[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.

[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.

[25]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, USA, 1998.

[26]

M. Flucher and M. Rumpf, Bernoulli's free-boundary problem, qualitative theory and numerical approximation, J. Reine Angew. Math., 486 (2003), 165-204.

[27]

P. Grisvard, Elliptic Problems in Non-smooth Domains, Pitman Publishing, Marshfield, Massachussetts, USA, 1985.

[28]

A. Friedman, Free boundary problems in science and technology, Notices of the AMS, 47 (2000), 854-861.

[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.

[30]

Z. GaoY. 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.

[31]

D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1983.

[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.

[33]

J. HaslingerK. ItoT. KozubekK. Kunisch and G. Peichl, On the shape derivative for problems of Bernoulli type, Interfaces Free Bound., 1 (2009), 317-330.

[34]

J. HaslingerT. KozubekK. 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.

[35]

J. HaslingerT. KozubekK. Kunisch and G. Peichl, Fictitious domain methods in shape optimization with applications in free-boundary problems, Comput. Optim. Appl., 26 (2003), 231-251.

[36]

J. HaslingerT. KozubekK. 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.

[37]

F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.

[38]

A. Henrot and A. Shangholian, Convexity of free boundaries with Bernoulli type boundary condition, Nonlinear Anal., 28 (1997), 815-823.

[39]

M. H. Imam, Three dimensional shape optimization, Int. J. Num. Meth. Engrg., 18 (1982), 661-673.

[40]

H. Kasumba, Shape optimization approaches to free-surface problems, Int. J. Numer. Meth. Fluids, 74 (2014), 818-845.

[41]

K. ItoK. Kunisch and G. Peichl, Variational approach to shape derivatives, ESAIM Control Optim. Calc. Var., 14 (2008), 517-539.

[42]

K. ItoK. Kunisch and G. Peichl, Variational approach to shape derivative for a class of Bernoulli problem, J. Math. Anal. Appl., 314 (2006), 126-149.

[43]

A. Laurain and H. Meftahi, Shape and parameter reconstruction for the Robin inverse problem, J. Inverse Ill-Posed Probl., 24 (2016), 643-662.

[44]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Berlin Heidelberg: Springer-Verlag, 1971.

[45]

H. Meftahi, Stability analysis in the inverse Robin transmission problem, Math. Methods Appl. Sci., 40 (2016), 2505-2521.

[46]

J. Neuberger, in Sobolev Gradients and Differential Equations (J-M. Morel and B. Teissier eds.), Lecture Notes in Mathematics. Springer: Berlin, 2010.

[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.

[48]

O. Pironneau and B. Mohammadi, Applied Shape Optimization in Fluid, Oxford University Press Inc: New York, 2001.

[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.

[50]

J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization, Springer, Berlin, Germany, 1991.

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, London, 1975.

[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.

[3]

H. Azegami, A solution to domain optimization problems, Trans of Japan Soc. of Mech. Engs., Ser. A, 60 (1994), 1479-1486 (in Japanese).

[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.

[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.

[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.

[7]

H. AzegamiS. Fukumoto and T. Aoyama, Shape optimization of continua using nurbs as basis functions, Struct. Multidiscipl. Optimiz., 47 (2013), 247-258.

[8]

H. AzegamiL. ZhouK. Umemura and N. Kondo, Shape optimization for a link mechanism, Struct. Multidiscipl. Optimiz., 48 (2013), 115-125.

[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.

[10]

J. Bacani, Methods of Shape Optimization in Free Boundary Problems, Ph. D. Thesis, Karl-Franzens-Universität Graz, Graz, Austria, 2013.

[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.

[12]

Z. Belhachmi and H. Meftahi, Shape sensitivity analysis for an interface problem via minimax differentiability, Appl.Math. Comput., 219 (2013), 6828-6842.

[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.

[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.

[15]

D. Chenais, On the existence of a solution in a domain identification problem, J. Math. Anal. Appl., (1975), 189-219.

[16]

R. Correa and A. Seeger, Directional derivative of a minimax function, Nonlinear Anal., 9 (1985), 13-22.

[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.

[18]

M. C. Delfour and J.-P. Zolésio, Shape sensitivity analysis via min max differentiability, SIAM J. Control Optim., 26 (1988), 834-862.

[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.

[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.

[21]

K. Eppler and H. Harbrecht, On a Kohn-Vogelius like formulation of free boundary problems, Comput. Optim. Appl., 52 (2012), 69-85.

[22]

K. Eppler and H. Harbrecht, Tracking Neumann data for stationary free boundary problems, SIAM J. Control Optim., 48 (2009), 2901-2916.

[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.

[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.

[25]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, USA, 1998.

[26]

M. Flucher and M. Rumpf, Bernoulli's free-boundary problem, qualitative theory and numerical approximation, J. Reine Angew. Math., 486 (2003), 165-204.

[27]

P. Grisvard, Elliptic Problems in Non-smooth Domains, Pitman Publishing, Marshfield, Massachussetts, USA, 1985.

[28]

A. Friedman, Free boundary problems in science and technology, Notices of the AMS, 47 (2000), 854-861.

[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.

[30]

Z. GaoY. 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.

[31]

D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1983.

[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.

[33]

J. HaslingerK. ItoT. KozubekK. Kunisch and G. Peichl, On the shape derivative for problems of Bernoulli type, Interfaces Free Bound., 1 (2009), 317-330.

[34]

J. HaslingerT. KozubekK. 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.

[35]

J. HaslingerT. KozubekK. Kunisch and G. Peichl, Fictitious domain methods in shape optimization with applications in free-boundary problems, Comput. Optim. Appl., 26 (2003), 231-251.

[36]

J. HaslingerT. KozubekK. 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.

[37]

F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.

[38]

A. Henrot and A. Shangholian, Convexity of free boundaries with Bernoulli type boundary condition, Nonlinear Anal., 28 (1997), 815-823.

[39]

M. H. Imam, Three dimensional shape optimization, Int. J. Num. Meth. Engrg., 18 (1982), 661-673.

[40]

H. Kasumba, Shape optimization approaches to free-surface problems, Int. J. Numer. Meth. Fluids, 74 (2014), 818-845.

[41]

K. ItoK. Kunisch and G. Peichl, Variational approach to shape derivatives, ESAIM Control Optim. Calc. Var., 14 (2008), 517-539.

[42]

K. ItoK. Kunisch and G. Peichl, Variational approach to shape derivative for a class of Bernoulli problem, J. Math. Anal. Appl., 314 (2006), 126-149.

[43]

A. Laurain and H. Meftahi, Shape and parameter reconstruction for the Robin inverse problem, J. Inverse Ill-Posed Probl., 24 (2016), 643-662.

[44]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Berlin Heidelberg: Springer-Verlag, 1971.

[45]

H. Meftahi, Stability analysis in the inverse Robin transmission problem, Math. Methods Appl. Sci., 40 (2016), 2505-2521.

[46]

J. Neuberger, in Sobolev Gradients and Differential Equations (J-M. Morel and B. Teissier eds.), Lecture Notes in Mathematics. Springer: Berlin, 2010.

[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.

[48]

O. Pironneau and B. Mohammadi, Applied Shape Optimization in Fluid, Oxford University Press Inc: New York, 2001.

[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.

[50]

J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization, Springer, Berlin, Germany, 1991.

Figure 1.  Initial and final shape of the annular domain $\Omega$
Figure 2.  Initial and final shape of the annular domain $\Omega$
Figure 3.  History of values of the cost functional $J$
Table 1.  Cost Values
Iter.Cost
(α = 0.001)
Cost
(α = 0.01)
1128.187510128.187510
20.198259950.36967179
30.061162570.00004442
40.000005120.00000004
Iter.Cost
(α = 0.001)
Cost
(α = 0.01)
1128.187510128.187510
20.198259950.36967179
30.061162570.00004442
40.000005120.00000004
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