# American Institute of Mathematical Sciences

December  2019, 8(4): 785-824. doi: 10.3934/eect.2019038

## A new energy-gap cost functional approach for the exterior Bernoulli free boundary problem

 Department of Complex Systems Science, Graduate School of Informatics, Nagoya University, A4-2 (780) Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan

* Corresponding author: J. F. T. Rabago

Received  August 2018 Revised  January 2019 Published  December 2019 Early access  June 2019

The solution to a free boundary problem of Bernoulli type, also known as Alt-Caffarelli problem, is studied via shape optimization techniques. In particular, a novel energy-gap cost functional approach with a state constraint consisting of a Robin condition is proposed as a shape optimization reformulation of the problem. Accordingly, the shape derivative of the cost is explicitly determined, and using the gradient information, a Lagrangian-like method is used to formulate an efficient boundary variation algorithm to numerically solve the minimization problem. The second order shape derivative of the cost is also computed, and through its characterization at the solution of the Bernoulli problem, the ill-posedness of the shape optimization formulation is proved. The analysis of the proposed formulation is completed by addressing the existence of optimal solution of the shape optimization problem and is accomplished by proving the continuity of the solution of the state problems with respect to the domain. The feasibility of the newly proposed method and its comparison with the classical energy-gap type cost functional approach is then presented through various numerical results. The numerical exploration issued in the study also includes results from a second-order optimization procedure based on a Newton-type method for resolving such minimization problem. This computational scheme put forward in the paper utilizes the Hessian information at the optimal solution and thus offers a state-of-the-art numerical approach for solving such free boundary problem via shape optimization setting.

Citation: Julius Fergy T. Rabago, Hideyuki Azegami. A new energy-gap cost functional approach for the exterior Bernoulli free boundary problem. Evolution Equations and Control Theory, 2019, 8 (4) : 785-824. doi: 10.3934/eect.2019038
##### References:
 [1] A. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine. Angew. Math., 325 (1981), 105-144.  doi: 10.1515/crll.1981.325.105. [2] H. Azegami, Second derivatives of cost functions and $H^1$ Newton method in shape optimization problems, Mathematical Analysis of Continuum Mechanics and Industrial Applications Ⅱ, in Proceedings of the International Conference CoMFoS16 (eds. P. van Meurs, M. Kimura and H. Notsu), vol. 30 of Mathematics for Industry, Springer, Singapore, (2017), 61–72. doi: 10.1007/978-981-10-6283-4_6. [3] H. Azegami, Shape Optimization Problems, Morikita Publishing Co., Ltd., Tokyo, 2016 (in Japanese). [4] H. Azegami, Solution of shape optimization problem and its application to product design, Mathematical Analysis of Continuum Mechanics and Industrial Applications, in Computer Aided Optimization Design of Structures IV, Structural Optimization (eds. H. Itou, M. Kimura, V. Chalupecký, K. Ohtsuka, D. Tagami and A. Takada), vol. 26 of Mathematics for Industry, Springer, Singapore, (2017), 83–98. doi: 10.1007/978-981-10-2633-1_6. [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 IV, Structural Optimization (eds. S. Hernandez and C. A. Brebbia), Computational Mechanics Publications, Southampton, (1997), 309–326. doi: 10.2495/OP970301. [6] H. Azegami and Z. Q. Wu, Domain optimization analysis in linear elastic problems: Approach using traction method, SME Int. J., Ser. A., 39 (1996), 272-278.  doi: 10.1299/jsmea1993.39.2_272. [7] H. Azegami, M. Shimoda, E. Katamine and Z. C. Wu, A domain optimization technique for elliptic boundary value problems, in Computer Aided Optimization Design of Structures IV, Structural Optimization (eds. S. Hernandez, M. El-Sayed and C. A. Brebbia), Computational Mechanics Publications, Southampton, (1995), 51–58. doi: 10.2495/OP950071. [8] H. Azegami, Solution to domain optimization problems, Trans. Jpn. Soc. Mech. Eng., Ser. A., 60 (1994), 1479–1486 (in Japanese). doi: 10.1299/kikaia.60.1479. [9] J. B. Bacani, Methods of Shape Optimization in Free Boundary Problems, Ph.D. Thesis, Karl-Franzens-Universität-Graz, 2013. [10] J. B. Bacani and G. Peichl, On the first-order shape derivative of the Kohn-Vogelius cost functional of the Bernoulli problem, Abstr. Appl. Anal., 2013 (2013), Art. ID 384320, 19pp. doi: 10.1155/2013/384320. [11] A. Ben Abda, F. Bouchon, G. Peichl, M. Sayeh and R. Touzani, A Dirichlet-Neumann cost functional approach for the Bernoulli problem, J. Eng. Math., 81 (2013), 157-176.  doi: 10.1007/s10665-012-9608-3. [12] A. Boulkhemair, A. Nachaoui and A. Chakib, A shape optimization approach for a class of free boundary problems of Bernoulli type, Appl. Math., 58 (2013), 205-221.  doi: 10.1007/s10492-013-0010-x. [13] A. Boulkhemair, A. Chakib and A. Nachaoui, Uniform trace theorem and application to shape optimization, Appl. Comput. Math., 7 (2008), 192-205. [14] A. Boulkhemair and A. Chakib, On the uniform Poincaré inequality, Commun. Partial Differ. Equations, 32 (2007), 1439-1447.  doi: 10.1080/03605300600910241. [15] D. Chenais, On the existence of a solution in a domain identification problem, J. Math. Anal. Appl., 52 (1975), 189-219.  doi: 10.1016/0022-247X(75)90091-8. [16] M. Dambrine, On variations of the shape Hessian and sufficient conditions for the stability of critical shapes, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A. Mat., 96 (2002), 95-121. [17] M. Dambrine and M. Pierre, About stability of equilibrium shapes, Model Math. Anal. Numer., 34 (2000), 811-834.  doi: 10.1051/m2an:2000105. [18] M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, 2nd edition, Adv. Des. Control 22, SIAM, Philadelphia, 2011. doi: 10.1137/1.9780898719826. [19] K. Eppler and H. Harbrecht, On a Kohn-Vogelius like formulation of free boundary problems, Comput. Optim. App., 52 (2012), 69-85.  doi: 10.1007/s10589-010-9345-3. [20] K. Eppler and H. Harbrecht, A regularized Newton method in electrical impedance tomography using shape Hessian information, Control Cybern., 34 (2005), 203-225. [21] K. Eppler, Boundary integral representations of second derivatives in shape optimization, Discuss. Math. Differ. Incl. Control. Optim., 20 (2000), 63-78.  doi: 10.7151/dmdico.1005. [22] K. Eppler, Optimal shape design for elliptic equations via BIE-methods, J. Appl. Math. Comput. Sci., 10 (2000), 487-516. [23] A. Fasano, Some free boundary problems with industrial applications, in Shape Optimization and Free Boundaries (eds. M. C. Delfour and G. Sabidussi), vol. 380 of NATO ASI Series (C: Mathematical and Physical Sciences), Springer, Dordrecht, (1992), 113–142. doi: 10.1007/978-94-011-2710-3_3. [24] M. Flucher and M. Rumpf, Bernoulli's free-boundary problem, qualitative theory and numerical approximation, J. Reine. Angew. Math., 486 (1997), 165-204.  doi: 10.1515/crll.1997.486.165. [25] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1977. doi: 10.1007/978-3-642-61798-0. [26] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Publishing, Marshfield, Massachusetts, 1985. doi: 10.1137/1.9781611972030. [27] J. Haslinger and R. A. E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation, and Computation, SIAM, Philadelphia, 2003. doi: 10.1137/1.9780898718690. [28] 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.  doi: 10.1016/j.jmaa.2003.10.038. [29] F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.  doi: 10.1515/jnum-2012-0013. [30] A. Henrot and M. Pierre, Shape Variation and Optimization: A Geometrical Analysis, Tracts in Mathematics 28, European Mathematical Society, Zürich, 2018. doi: 10.4171/178. [31] K. Ito, K. Kunisch and G. Peichl, Variational approach to shape derivatives, ESAIM Control Optim. Calc. Var., 14 (2008), 517-539.  doi: 10.1051/cocv:2008002. [32] T. Kashiwabara, C. M. Colciago, L. Dedè and A. Quarteroni, Well-posedness, regularity, and convergence analysis of the finite element approximation of a generalized Robin boundary value problem, SIAM J. Numer. Anal., 53 (2015), 105-126.  doi: 10.1137/140954477. [33] R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Commun. Pure Appl., 37 (1984), 289-298.  doi: 10.1002/cpa.3160370302. [34] R. Kress, On Trefftz' integral equation for the Bernoulli free boundary value problem, Numer. Math., 136 (2017), 503-522.  doi: 10.1007/s00211-016-0847-5. [35] J. Nečas, Direct Methods in the Theory of Elliptic Equations, Corrected 2nd edition, Monographs and Studies in Mathematics, Springer, Berlin, Heidelberg, 2012. doi: 10.1007/978-3-642-10455-8. [36] J. W. Neuberger, Sobolev Gradients and Differential Equations, Springer-Verlag, Berlin, 1997. doi: 10.1007/BFb0092831. [37] A. Novruzi and M. Pierre, Structure of shape derivatives, J. Evol. Equ., 2 (2002), 365-382.  doi: 10.1007/s00028-002-8093-y. [38] A. Novruzi and J.-R. Roche, Newton's method in shape optimisation: A three-dimensional case, BIT Numer. Math., 40 (2000), 102-120.  doi: 10.1023/A:1022370419231. [39] S. Osher and J. Sethian, Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys., 79 (1988), 12-49.  doi: 10.1016/0021-9991(88)90002-2. [40] A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, vol. 37 of Texts in Applied Mathematics, 2nd edition, Springer, Berlin, 2007. doi: 10.1007/b98885. [41] J. F. T. Rabago and H. Azegami, An improved shape optimization formulation of the Bernoulli problem by tracking the Neumann data, J. Eng. Math., (2019), to appear. [42] J. F. T. Rabago and J. B. Bacani, Shape optimization approach to the Bernoulli problem: A Lagrangian formulation, IAENG Int. J. Appl. Math., 47 (2017), 417-424. [43] J. F. T. Rabago and J. B. Bacani, Shape optimization approach for solving the Bernoulli problem by tracking the Neumann data: a Lagrangian formulation, Commun. Pur. Appl. Anal., 17 (2018), 2683-2702.  doi: 10.3934/cpaa.2018127. [44] J. Simon, Second variation for domain optimization problems, in Control and Estimation of Distributed Parameter Systems (eds. F. Kappel, K. Kunisch and W. Schappacher), International Series of Numerical Mathematics, no 91. Birkhäuser, (1989), 361–378. [45] J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization, in Introduction to Shape Optimization, vol. 16 of Springer Series in Computational Mathematics, Springer, Berlin, Heidelberg, 1992. doi: 10.1007/978-3-642-58106-9_1. [46] T. Tiihonen, Shape optimization and trial methods for free boundary problems, RAIRO Modél. Math. Anal. Numér., 31 (1997), 805-825.  doi: 10.1051/m2an/1997310708051.

show all references

##### References:
 [1] A. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine. Angew. Math., 325 (1981), 105-144.  doi: 10.1515/crll.1981.325.105. [2] H. Azegami, Second derivatives of cost functions and $H^1$ Newton method in shape optimization problems, Mathematical Analysis of Continuum Mechanics and Industrial Applications Ⅱ, in Proceedings of the International Conference CoMFoS16 (eds. P. van Meurs, M. Kimura and H. Notsu), vol. 30 of Mathematics for Industry, Springer, Singapore, (2017), 61–72. doi: 10.1007/978-981-10-6283-4_6. [3] H. Azegami, Shape Optimization Problems, Morikita Publishing Co., Ltd., Tokyo, 2016 (in Japanese). [4] H. Azegami, Solution of shape optimization problem and its application to product design, Mathematical Analysis of Continuum Mechanics and Industrial Applications, in Computer Aided Optimization Design of Structures IV, Structural Optimization (eds. H. Itou, M. Kimura, V. Chalupecký, K. Ohtsuka, D. Tagami and A. Takada), vol. 26 of Mathematics for Industry, Springer, Singapore, (2017), 83–98. doi: 10.1007/978-981-10-2633-1_6. [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 IV, Structural Optimization (eds. S. Hernandez and C. A. Brebbia), Computational Mechanics Publications, Southampton, (1997), 309–326. doi: 10.2495/OP970301. [6] H. Azegami and Z. Q. Wu, Domain optimization analysis in linear elastic problems: Approach using traction method, SME Int. J., Ser. A., 39 (1996), 272-278.  doi: 10.1299/jsmea1993.39.2_272. [7] H. Azegami, M. Shimoda, E. Katamine and Z. C. Wu, A domain optimization technique for elliptic boundary value problems, in Computer Aided Optimization Design of Structures IV, Structural Optimization (eds. S. Hernandez, M. El-Sayed and C. A. Brebbia), Computational Mechanics Publications, Southampton, (1995), 51–58. doi: 10.2495/OP950071. [8] H. Azegami, Solution to domain optimization problems, Trans. Jpn. Soc. Mech. Eng., Ser. A., 60 (1994), 1479–1486 (in Japanese). doi: 10.1299/kikaia.60.1479. [9] J. B. Bacani, Methods of Shape Optimization in Free Boundary Problems, Ph.D. Thesis, Karl-Franzens-Universität-Graz, 2013. [10] J. B. Bacani and G. Peichl, On the first-order shape derivative of the Kohn-Vogelius cost functional of the Bernoulli problem, Abstr. Appl. Anal., 2013 (2013), Art. ID 384320, 19pp. doi: 10.1155/2013/384320. [11] A. Ben Abda, F. Bouchon, G. Peichl, M. Sayeh and R. Touzani, A Dirichlet-Neumann cost functional approach for the Bernoulli problem, J. Eng. Math., 81 (2013), 157-176.  doi: 10.1007/s10665-012-9608-3. [12] A. Boulkhemair, A. Nachaoui and A. Chakib, A shape optimization approach for a class of free boundary problems of Bernoulli type, Appl. Math., 58 (2013), 205-221.  doi: 10.1007/s10492-013-0010-x. [13] A. Boulkhemair, A. Chakib and A. Nachaoui, Uniform trace theorem and application to shape optimization, Appl. Comput. Math., 7 (2008), 192-205. [14] A. Boulkhemair and A. Chakib, On the uniform Poincaré inequality, Commun. Partial Differ. Equations, 32 (2007), 1439-1447.  doi: 10.1080/03605300600910241. [15] D. Chenais, On the existence of a solution in a domain identification problem, J. Math. Anal. Appl., 52 (1975), 189-219.  doi: 10.1016/0022-247X(75)90091-8. [16] M. Dambrine, On variations of the shape Hessian and sufficient conditions for the stability of critical shapes, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A. Mat., 96 (2002), 95-121. [17] M. Dambrine and M. Pierre, About stability of equilibrium shapes, Model Math. Anal. Numer., 34 (2000), 811-834.  doi: 10.1051/m2an:2000105. [18] M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, 2nd edition, Adv. Des. Control 22, SIAM, Philadelphia, 2011. doi: 10.1137/1.9780898719826. [19] K. Eppler and H. Harbrecht, On a Kohn-Vogelius like formulation of free boundary problems, Comput. Optim. App., 52 (2012), 69-85.  doi: 10.1007/s10589-010-9345-3. [20] K. Eppler and H. Harbrecht, A regularized Newton method in electrical impedance tomography using shape Hessian information, Control Cybern., 34 (2005), 203-225. [21] K. Eppler, Boundary integral representations of second derivatives in shape optimization, Discuss. Math. Differ. Incl. Control. Optim., 20 (2000), 63-78.  doi: 10.7151/dmdico.1005. [22] K. Eppler, Optimal shape design for elliptic equations via BIE-methods, J. Appl. Math. Comput. Sci., 10 (2000), 487-516. [23] A. Fasano, Some free boundary problems with industrial applications, in Shape Optimization and Free Boundaries (eds. M. C. Delfour and G. Sabidussi), vol. 380 of NATO ASI Series (C: Mathematical and Physical Sciences), Springer, Dordrecht, (1992), 113–142. doi: 10.1007/978-94-011-2710-3_3. [24] M. Flucher and M. Rumpf, Bernoulli's free-boundary problem, qualitative theory and numerical approximation, J. Reine. Angew. Math., 486 (1997), 165-204.  doi: 10.1515/crll.1997.486.165. [25] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1977. doi: 10.1007/978-3-642-61798-0. [26] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Publishing, Marshfield, Massachusetts, 1985. doi: 10.1137/1.9781611972030. [27] J. Haslinger and R. A. E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation, and Computation, SIAM, Philadelphia, 2003. doi: 10.1137/1.9780898718690. [28] 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.  doi: 10.1016/j.jmaa.2003.10.038. [29] F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.  doi: 10.1515/jnum-2012-0013. [30] A. Henrot and M. Pierre, Shape Variation and Optimization: A Geometrical Analysis, Tracts in Mathematics 28, European Mathematical Society, Zürich, 2018. doi: 10.4171/178. [31] K. Ito, K. Kunisch and G. Peichl, Variational approach to shape derivatives, ESAIM Control Optim. Calc. Var., 14 (2008), 517-539.  doi: 10.1051/cocv:2008002. [32] T. Kashiwabara, C. M. Colciago, L. Dedè and A. Quarteroni, Well-posedness, regularity, and convergence analysis of the finite element approximation of a generalized Robin boundary value problem, SIAM J. Numer. Anal., 53 (2015), 105-126.  doi: 10.1137/140954477. [33] R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Commun. Pure Appl., 37 (1984), 289-298.  doi: 10.1002/cpa.3160370302. [34] R. Kress, On Trefftz' integral equation for the Bernoulli free boundary value problem, Numer. Math., 136 (2017), 503-522.  doi: 10.1007/s00211-016-0847-5. [35] J. Nečas, Direct Methods in the Theory of Elliptic Equations, Corrected 2nd edition, Monographs and Studies in Mathematics, Springer, Berlin, Heidelberg, 2012. doi: 10.1007/978-3-642-10455-8. [36] J. W. Neuberger, Sobolev Gradients and Differential Equations, Springer-Verlag, Berlin, 1997. doi: 10.1007/BFb0092831. [37] A. Novruzi and M. Pierre, Structure of shape derivatives, J. Evol. Equ., 2 (2002), 365-382.  doi: 10.1007/s00028-002-8093-y. [38] A. Novruzi and J.-R. Roche, Newton's method in shape optimisation: A three-dimensional case, BIT Numer. Math., 40 (2000), 102-120.  doi: 10.1023/A:1022370419231. [39] S. Osher and J. Sethian, Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys., 79 (1988), 12-49.  doi: 10.1016/0021-9991(88)90002-2. [40] A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, vol. 37 of Texts in Applied Mathematics, 2nd edition, Springer, Berlin, 2007. doi: 10.1007/b98885. [41] J. F. T. Rabago and H. Azegami, An improved shape optimization formulation of the Bernoulli problem by tracking the Neumann data, J. Eng. Math., (2019), to appear. [42] J. F. T. Rabago and J. B. Bacani, Shape optimization approach to the Bernoulli problem: A Lagrangian formulation, IAENG Int. J. Appl. Math., 47 (2017), 417-424. [43] J. F. T. Rabago and J. B. Bacani, Shape optimization approach for solving the Bernoulli problem by tracking the Neumann data: a Lagrangian formulation, Commun. Pur. Appl. Anal., 17 (2018), 2683-2702.  doi: 10.3934/cpaa.2018127. [44] J. Simon, Second variation for domain optimization problems, in Control and Estimation of Distributed Parameter Systems (eds. F. Kappel, K. Kunisch and W. Schappacher), International Series of Numerical Mathematics, no 91. Birkhäuser, (1989), 361–378. [45] J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization, in Introduction to Shape Optimization, vol. 16 of Springer Series in Computational Mathematics, Springer, Berlin, Heidelberg, 1992. doi: 10.1007/978-3-642-58106-9_1. [46] T. Tiihonen, Shape optimization and trial methods for free boundary problems, RAIRO Modél. Math. Anal. Numér., 31 (1997), 805-825.  doi: 10.1051/m2an/1997310708051.
Initial (left) and final (right) free boundaries for Example 1 with $\alpha = 0.1$ in (57)
(a)-(b): Respective histories of cost values and Hausdorff distances via first-order method with $\alpha = 0.1$ in (57) and varying initial free boundary $\Sigma_0^i$, $i = 1, 2, 3$; (c)-(d): respective histories of cost values and Hausdorff distances via second-order method with $\alpha = 0.1$ in (57) and different initial free boundary $\Sigma_0^i$, $i = 1, 2, 3$
Blue solid lines: Optimal free boundaries for Example 2 when $\lambda = -10, -9, \ldots, -1$ (the outermost boundary corresponds to $\lambda = -1$ and the innermost boundary to $\lambda = -10$); dashed-dot magenta line: initial guess for the free boundary
Results of Example 2 for $\lambda = -9$ when $\eta = 10^{-6}$ in the stopping condition (58) and $\alpha = 0.99$ in (57)
(a): Histories of descent step sizes for the proposed and classical formulations (with almost equal initial step size $t_0$ for the two formulations); (b): optimal free boundaries obtained when $\lambda = -9, -4, -1$ in Example 2 using the proposed and classical formulations with $\eta = 10^{-4}$ in the stopping condition (58)
(a)-(b): histories of free boundaries obtained through the proposed formulation with initial guess $\Sigma_0^1$ and $\Sigma_0^3$, respectively, where $\eta = 10^{-6}$ in (58); (c)-(d): Histories of free boundaries obtained via the classical Kohn-Vogelius formulation with initial guess $\Sigma_0^1$ and $\Sigma_0^3$, respectively, where $\eta = 10^{-4}$ in (58)
(a)-(b): corresponding histories of curvatures of the free boundaries obtained through the proposed formulation with initial guess $\Sigma_0^1$ and $\Sigma_0^3$, respectively, shown in Figure 6a-6b; (c)-(d): Corresponding histories of curvatures of the free boundaries obtained via the classical Kohn-Vogelius formulation with initial guess $\Sigma_0^1$ and $\Sigma_0^3$, respectively, shown in Figure 6c-6d
Blue solid lines: Optimal free boundaries for Example 3 when $\lambda = -10, -9, \ldots, -1$ (the outermost boundary corresponds to $\lambda = -1$ and the innermost boundary to $\lambda = -10$); dashed-dot magenta line: initial guess for the free boundary
Results of Example 3 when $\lambda = -10, -4, -1$ for both of the proposed and classical formulations
(a)-(b): histories of free boundaries obtained through the proposed formulation with initial guess $C(\boldsymbol{0}, 0.6)$ and $\Upsilon$, respectively, where $\eta = 10^{-6}$ in (58); (c)-(d): histories of free boundaries obtained via the classical Kohn-Vogelius formulation with initial guess $C(\boldsymbol{0}, 0.6)$ and $\Upsilon$, respectively, where $\eta = 10^{-4}$ in (58)
(a)-(b): Corresponding histories of curvatures of the free boundaries obtained through the proposed formulation with initial guess $\Sigma_0^1$ and $\Sigma_0^3$ shown in Figure 10a-10b, respectively; (c)-(d): corresponding histories of curvatures of the free boundaries obtained via the classical Kohn-Vogelius formulation with initial guess $\Sigma_0^1$ and $\Sigma_0^3$ shown in Figure 10c-10d, respectively
Convergence test toward exact solution using the proposed formulation via the modified $H^1$-gradient method with initial free boundaries $\Sigma_0^i$, $i = 1, 2, 3$, and $\alpha = 0.10, 0.50, 0.99$ in (57)
 $\Sigma_0^i$ $\alpha$ cost ${\rm d}_{\rm H}(\Sigma^\ast, \Sigma^i_f)$ $\bar{R}$ $|R - \bar{R}|/R$ iter. cpu time $\Sigma_0^1$ 0.10 $2.85 \times 10^{-5}$ $0.005072$ $0.500888$ 0.001776 72 115 sec 0.50 $2.32 \times 10^{-7}$ $0.004983$ $0.500002$ 0.000004 17 26 sec 0.99 $8.55 \times 10^{-8}$ $0.004984$ $0.499865$ 0.000270 8 12 sec $\Sigma_0^2$ 0.10 $1.77 \times 10^{-5}$ $0.005044$ $0.499343$ 0.001314 70 103 sec 0.50 $9.26 \times 10^{-7}$ $0.005003$ $0.499878$ 0.000244 16 28 sec 0.99 $3.91 \times 10^{-9}$ $0.004998$ $0.499956$ 0.000088 7 14 sec $\Sigma_0^3$ 0.10 $1.65 \times 10^{-5}$ $0.005887$ $0.500051$ 0.000102 76 122 sec 0.50 $6.64 \times 10^{-7}$ $0.004991$ $0.500002$ 0.000004 19 29 sec 0.99 $8.77 \times 10^{-7}$ $0.005001$ $0.499993$ 0.000014 9 13 sec
 $\Sigma_0^i$ $\alpha$ cost ${\rm d}_{\rm H}(\Sigma^\ast, \Sigma^i_f)$ $\bar{R}$ $|R - \bar{R}|/R$ iter. cpu time $\Sigma_0^1$ 0.10 $2.85 \times 10^{-5}$ $0.005072$ $0.500888$ 0.001776 72 115 sec 0.50 $2.32 \times 10^{-7}$ $0.004983$ $0.500002$ 0.000004 17 26 sec 0.99 $8.55 \times 10^{-8}$ $0.004984$ $0.499865$ 0.000270 8 12 sec $\Sigma_0^2$ 0.10 $1.77 \times 10^{-5}$ $0.005044$ $0.499343$ 0.001314 70 103 sec 0.50 $9.26 \times 10^{-7}$ $0.005003$ $0.499878$ 0.000244 16 28 sec 0.99 $3.91 \times 10^{-9}$ $0.004998$ $0.499956$ 0.000088 7 14 sec $\Sigma_0^3$ 0.10 $1.65 \times 10^{-5}$ $0.005887$ $0.500051$ 0.000102 76 122 sec 0.50 $6.64 \times 10^{-7}$ $0.004991$ $0.500002$ 0.000004 19 29 sec 0.99 $8.77 \times 10^{-7}$ $0.005001$ $0.499993$ 0.000014 9 13 sec
Convergence test toward exact solution using the proposed formulation via the modified $H^1$-Newton method with $\alpha = 0.1$ and different initial free boundary $\Sigma_0^i$, $i = 1, 2, 3$
 $\alpha$ $\Sigma_0^i$ cost ${\rm d}_{\rm H}(\Sigma^\ast, \Sigma^i_f)$ $\bar{R}$ $|R - \bar{R}|/R$ iter. cpu time $0.1$ $\Sigma_0^1$ $6.17 \times 10^{-7}$ $0.005007$ $0.500139$ 0.000278 7 25 sec $\Sigma_0^2$ $1.57 \times 10^{-8}$ $0.005003$ $0.500013$ 0.000026 8 41 sec $\Sigma_0^3$ $4.47 \times 10^{-6}$ $0.005130$ $0.500101$ 0.000202 14 35 sec
 $\alpha$ $\Sigma_0^i$ cost ${\rm d}_{\rm H}(\Sigma^\ast, \Sigma^i_f)$ $\bar{R}$ $|R - \bar{R}|/R$ iter. cpu time $0.1$ $\Sigma_0^1$ $6.17 \times 10^{-7}$ $0.005007$ $0.500139$ 0.000278 7 25 sec $\Sigma_0^2$ $1.57 \times 10^{-8}$ $0.005003$ $0.500013$ 0.000026 8 41 sec $\Sigma_0^3$ $4.47 \times 10^{-6}$ $0.005130$ $0.500101$ 0.000202 14 35 sec
Summary of computational results for an L-shaped fixed boundary $\Gamma = \partial S$ with $\lambda = -10, -9, \ldots, -1$ where $\alpha = 0.99$ in (57) and $\eta = 10^{-6}$ in the stopping condition (58)
 $\lambda$ formulation $t_0$ cost iteration cpu time $-10$ proposed $0.228150$ $1.37 \times 10^{-6}$ $14$ $14$ sec classical $0.767890$ $0.000137$ $19$ $55$ sec $-9$ proposed $0.221636$ $6.25 \times 10^{-7}$ $13$ $16$ sec classical $0.738627$ $3.11 \times 10^{-5}$ $25$ $96$ sec $-8$ proposed $0.213501$ $7.94 \times 10^{-7}$ $12$ $14$ sec classical $0.702851$ $7.17 \times 10^{-5}$ $19$ $57$ sec $-7$ proposed $0.203058$ $1.23 \times 10^{-6}$ $10$ $13$ sec classical $0.658125$ $0.000628$ $12$ $34$ sec $-6$ proposed $0.189163$ $8.27 \times 10^{-7}$ $10$ $14$ sec classical $0.600640$ $0.000190$ $13$ $34$ sec $-5$ proposed $0.169783$ $2.61 \times 10^{-7}$ $10$ $16$ sec classical $0.524113$ $0.000948$ $18$ $54$ sec $-4$ proposed $0.140942$ $2.04 \times 10^{-7}$ $10$ $17$ sec classical $0.417590$ $0.000186$ $8$ $24$ sec $-3$ proposed $0.094111$ $5.68 \times 10^{-7}$ $9$ $14$ sec classical $0.262124$ $4.95 \times 10^{-5}$ $11$ $29$ sec $-2$ proposed $0.039805$ $2.37 \times 10^{-7}$ $10$ $17$ sec classical $0.120956$ $6.48 \times 10^{-6}$ $10$ $27$ sec $-1$ proposed $0.312388$ $1.08 \times 10^{-6}$ $13$ $25$ sec classical $0.615256$ $5.69 \times 10^{-7}$ $9$ $24$ sec
 $\lambda$ formulation $t_0$ cost iteration cpu time $-10$ proposed $0.228150$ $1.37 \times 10^{-6}$ $14$ $14$ sec classical $0.767890$ $0.000137$ $19$ $55$ sec $-9$ proposed $0.221636$ $6.25 \times 10^{-7}$ $13$ $16$ sec classical $0.738627$ $3.11 \times 10^{-5}$ $25$ $96$ sec $-8$ proposed $0.213501$ $7.94 \times 10^{-7}$ $12$ $14$ sec classical $0.702851$ $7.17 \times 10^{-5}$ $19$ $57$ sec $-7$ proposed $0.203058$ $1.23 \times 10^{-6}$ $10$ $13$ sec classical $0.658125$ $0.000628$ $12$ $34$ sec $-6$ proposed $0.189163$ $8.27 \times 10^{-7}$ $10$ $14$ sec classical $0.600640$ $0.000190$ $13$ $34$ sec $-5$ proposed $0.169783$ $2.61 \times 10^{-7}$ $10$ $16$ sec classical $0.524113$ $0.000948$ $18$ $54$ sec $-4$ proposed $0.140942$ $2.04 \times 10^{-7}$ $10$ $17$ sec classical $0.417590$ $0.000186$ $8$ $24$ sec $-3$ proposed $0.094111$ $5.68 \times 10^{-7}$ $9$ $14$ sec classical $0.262124$ $4.95 \times 10^{-5}$ $11$ $29$ sec $-2$ proposed $0.039805$ $2.37 \times 10^{-7}$ $10$ $17$ sec classical $0.120956$ $6.48 \times 10^{-6}$ $10$ $27$ sec $-1$ proposed $0.312388$ $1.08 \times 10^{-6}$ $13$ $25$ sec classical $0.615256$ $5.69 \times 10^{-7}$ $9$ $24$ sec
Computational results obtained via the classical formulation with $\eta = 10^{-6}$ in the stopping condition (58) for an L-shaped fixed boundary $\Gamma = \partial S$ when $\lambda = -9, -4, -1$ for different values of $\alpha$ in (57)
 $\lambda$ $i$ $\alpha_i$ $t_0$ cost iteration cpu time $-9$ $1$ $0.2970642705$ $0.2216361137$ $1.34 \times 10^{-5}$ $22$ $74$ sec $2$ $0.2970642710$ $0.2216361144$ $3.70 \times 10^{-5}$ $18$ $102$ sec $3$ $0.2970642715$ $0.2216361144$ $6.40 \times 10^{-5}$ $17$ $32$ sec $-4$ $1$ $0.334138300$ $0.1409423055$ $1.03 \times 10^{-5}$ $11$ $23$ sec $2$ $0.334138305$ $0.1409423076$ $4.57 \times 10^{-6}$ $12$ $26$ sec $3$ $0.334138310$ $0.1409423098$ $4.92 \times 10^{-5}$ $14$ $40$ sec $-1$ $1$ $0.502658435$ $0.3123875250$ $1.34 \times 10^{-6}$ $14$ $34$ sec $2$ $0.502658440$ $0.3123875282$ $1.17 \times 10^{-6}$ $14$ $35$ sec $3$ $0.502658445$ $0.3123875313$ $1.24 \times 10^{-6}$ $14$ $34$ sec
 $\lambda$ $i$ $\alpha_i$ $t_0$ cost iteration cpu time $-9$ $1$ $0.2970642705$ $0.2216361137$ $1.34 \times 10^{-5}$ $22$ $74$ sec $2$ $0.2970642710$ $0.2216361144$ $3.70 \times 10^{-5}$ $18$ $102$ sec $3$ $0.2970642715$ $0.2216361144$ $6.40 \times 10^{-5}$ $17$ $32$ sec $-4$ $1$ $0.334138300$ $0.1409423055$ $1.03 \times 10^{-5}$ $11$ $23$ sec $2$ $0.334138305$ $0.1409423076$ $4.57 \times 10^{-6}$ $12$ $26$ sec $3$ $0.334138310$ $0.1409423098$ $4.92 \times 10^{-5}$ $14$ $40$ sec $-1$ $1$ $0.502658435$ $0.3123875250$ $1.34 \times 10^{-6}$ $14$ $34$ sec $2$ $0.502658440$ $0.3123875282$ $1.17 \times 10^{-6}$ $14$ $35$ sec $3$ $0.502658445$ $0.3123875313$ $1.24 \times 10^{-6}$ $14$ $34$ sec
Comparison of computational results obtained through the proposed and classical formulations with $\eta = 10^{-4}$ in (58) for an L-shaped fixed boundary $\Gamma = \partial S$ when $\lambda = -9, -4, -1$ with almost the same initial step size $t_0$ for both formulations
 $\lambda$ formulation $\alpha$ $t_0$ cost iteration cpu time $-9$ proposed $0.990000000$ $0.2216361144$ $1.55 \times 10^{-5}$ $9$ $10$ sec classical $0.297064271$ $0.2216361144$ $14.2 \times 10^{-5}$ $12$ $16$ sec $-4$ proposed $0.990000000$ $0.1409423086$ $6.37 \times 10^{-6}$ $7$ $11$ sec classical $0.334138305$ $0.1409423076$ $4.57 \times 10^{-6}$ $7$ $11$ sec $-1$ proposed $0.990000000$ $0.3123875327$ $4.39 \times 10^{-5}$ $8$ $17$ sec classical $0.502658440$ $0.3123875282$ $5.51 \times 10^{-5}$ $9$ $24$ sec
 $\lambda$ formulation $\alpha$ $t_0$ cost iteration cpu time $-9$ proposed $0.990000000$ $0.2216361144$ $1.55 \times 10^{-5}$ $9$ $10$ sec classical $0.297064271$ $0.2216361144$ $14.2 \times 10^{-5}$ $12$ $16$ sec $-4$ proposed $0.990000000$ $0.1409423086$ $6.37 \times 10^{-6}$ $7$ $11$ sec classical $0.334138305$ $0.1409423076$ $4.57 \times 10^{-6}$ $7$ $11$ sec $-1$ proposed $0.990000000$ $0.3123875327$ $4.39 \times 10^{-5}$ $8$ $17$ sec classical $0.502658440$ $0.3123875282$ $5.51 \times 10^{-5}$ $9$ $24$ sec
Comparison of computational results obtained through the proposed and classical formulations for an L-shaped fixed boundary $\Gamma = \partial S$ when $\lambda = -10$ with almost the same initial step size $t_0$ for both formulations
 $\Sigma_0^i$ formulation $\eta$ cost ${\rm d}_{\rm H}(\Sigma^{\rm in}, \Sigma^i_f)$ iteration cpu time $\Sigma_0^1$ proposed $10^{-4}$ $2.93 \times 10^{-5}$ $0.023850$ $9$ $10$ sec $10^{-5}$ $4.46 \times 10^{-6}$ $0.008586$ $12$ $13$ sec $10^{-6}$ $1.38 \times 10^{-6}$ $0.008586$ $14$ $14$ sec classical $10^{-4}$ $0.001627$ $0.008435$ $9$ $14$ sec $10^{-5}$ $0.001627$ $0.010512$ $9$ $14$ sec $10^{-6}$ $0.000224$ $0.007484$ $15$ $30$ sec $\Sigma_0^3$ proposed $10^{-4}$ $5.69 \times 10^{-5}$ $0.026360$ $8$ $9$ sec $10^{-5}$ $1.22 \times 10^{-5}$ $0.008565$ $10$ $10$ sec $10^{-6}$ $9.47 \times 10^{-7}$ $0.007675$ $14$ $14$ sec classical $10^{-4}$ $0.000644$ $0.008637$ $9$ $15$ sec $10^{-5}$ $0.000062$ $0.007394$ $15$ $30$ sec $10^{-6}$ $0.000027$ $0.007394$ $19$ $114$ sec
 $\Sigma_0^i$ formulation $\eta$ cost ${\rm d}_{\rm H}(\Sigma^{\rm in}, \Sigma^i_f)$ iteration cpu time $\Sigma_0^1$ proposed $10^{-4}$ $2.93 \times 10^{-5}$ $0.023850$ $9$ $10$ sec $10^{-5}$ $4.46 \times 10^{-6}$ $0.008586$ $12$ $13$ sec $10^{-6}$ $1.38 \times 10^{-6}$ $0.008586$ $14$ $14$ sec classical $10^{-4}$ $0.001627$ $0.008435$ $9$ $14$ sec $10^{-5}$ $0.001627$ $0.010512$ $9$ $14$ sec $10^{-6}$ $0.000224$ $0.007484$ $15$ $30$ sec $\Sigma_0^3$ proposed $10^{-4}$ $5.69 \times 10^{-5}$ $0.026360$ $8$ $9$ sec $10^{-5}$ $1.22 \times 10^{-5}$ $0.008565$ $10$ $10$ sec $10^{-6}$ $9.47 \times 10^{-7}$ $0.007675$ $14$ $14$ sec classical $10^{-4}$ $0.000644$ $0.008637$ $9$ $15$ sec $10^{-5}$ $0.000062$ $0.007394$ $15$ $30$ sec $10^{-6}$ $0.000027$ $0.007394$ $19$ $114$ sec
Computational results obtained through the classical formulation with $\eta = 10^{-4}$ in (58) for an L-shaped fixed boundary $\Gamma = \partial S$ for $\lambda = -10, -8, -7, -6, -5, -3, -2$ with almost the same initial step size $t_0$ with respect to that of the proposed formulation shown in Table 3
 $\lambda$ $\alpha$ $t_0$ cost iteration cpu time $-10$ $0.29414181$ $0.22815001$ $0.001628$ $9$ $14$ sec $-8$ $0.30072761$ $0.21350176$ $0.000119$ $12$ $17$ sec $-7$ $0.30545458$ $0.20305801$ $0.001727$ $10$ $17$ sec $-6$ $0.31178679$ $0.18916314$ $0.000137$ $8$ $13$ sec $-5$ $0.32070413$ $0.16978307$ $0.000925$ $6$ $10$ sec $-3$ $0.35544029$ $0.09411051$ $7.89 \times 10^{-5}$ $6$ $14$ sec $-2$ $0.32579718$ $0.03980505$ $2.08 \times 10^{-5}$ $7$ $14$ sec
 $\lambda$ $\alpha$ $t_0$ cost iteration cpu time $-10$ $0.29414181$ $0.22815001$ $0.001628$ $9$ $14$ sec $-8$ $0.30072761$ $0.21350176$ $0.000119$ $12$ $17$ sec $-7$ $0.30545458$ $0.20305801$ $0.001727$ $10$ $17$ sec $-6$ $0.31178679$ $0.18916314$ $0.000137$ $8$ $13$ sec $-5$ $0.32070413$ $0.16978307$ $0.000925$ $6$ $10$ sec $-3$ $0.35544029$ $0.09411051$ $7.89 \times 10^{-5}$ $6$ $14$ sec $-2$ $0.32579718$ $0.03980505$ $2.08 \times 10^{-5}$ $7$ $14$ sec
Means and standard deviations (std) of the number of iterations, computing time and computing time per iteration for the proposed formulation with $\eta = 10^{-6}$ and classical formulation with $\eta = 10^{-4}$ in (58)
 formulation iteration cpu time $\frac{\rm cpu\ time}{\rm iteration}$ mean std mean std mean std proposed $\approx 11 (11.1)$ $\approx 2 (1.73)$ $16$ $3.46$ $1.46$ $0.29$ classical $\approx 9 (8.6)$ $\approx 2 (2.22)$ $15$ $3.92$ $1.77$ $0.42$
 formulation iteration cpu time $\frac{\rm cpu\ time}{\rm iteration}$ mean std mean std mean std proposed $\approx 11 (11.1)$ $\approx 2 (1.73)$ $16$ $3.46$ $1.46$ $0.29$ classical $\approx 9 (8.6)$ $\approx 2 (2.22)$ $15$ $3.92$ $1.77$ $0.42$
Summary of computational results of Example 3 for $\lambda = -10, -9, \ldots, -1$ where the highlighted rows correspond to the results due to the proposed formulation
 $\lambda$ $\alpha$ $t_0$ cost iter cpu time $-10$ $0.990000000$ $0.337420581$ $1.33 \times 10^{-7}$ $14$ $14$ sec $0.386646216$ $0.337420581$ $0.001223$ $15$ $24$ sec $-9$ $0.990000000$ $0.331971040$ $5.73 \times 10^{-7}$ $13$ $16$ sec $0.388463867$ $0.331971040$ $0.000473$ $11$ $20$ sec $-8$ $0.990000000$ $0.325160189$ $5.07 \times 10^{-7}$ $13$ $14$ sec $0.390736229$ $0.325160189$ $0.000582$ $10$ $17$ sec $-7$ $0.990000000$ $0.316405255$ $3.36 \times 10^{-7}$ $12$ $18$ sec $0.393657966$ $0.316405255$ $0.000745$ $10$ $16$ sec $-6$ $0.990000000$ $0.304735585$ $3.79 \times 10^{-7}$ $11$ $14$ sec $0.397552887$ $0.304735585$ $9.84 \times 10^{-5}$ $12$ $20$ sec $-5$ $0.990000000$ $0.288405794$ $3.33 \times 10^{-7}$ $11$ $13$ sec $0.403001262$ $0.288405794$ $4.38 \times 10^{-5}$ $13$ $18$ sec $-4$ $0.990000000$ $0.263931464$ $2.09 \times 10^{-7}$ $11$ $14$ sec $0.411150343$ $0.263931464$ $2.25 \times 10^{-5}$ $9$ $16$ sec $-3$ $0.990000000$ $0.223215311$ $1.45 \times 10^{-7}$ $11$ $13$ sec $0.424571657$ $0.223215311$ $3.88 \times 10^{-5}$ $7$ $11$ sec $-2$ $0.990000000$ $0.142355886$ $1.70 \times 10^{-7}$ $10$ $17$ sec $0.448686986$ $0.142355886$ $1.30 \times 10^{-5}$ $7$ $13$ sec $-1$ $0.990000000$ $0.111335863$ $9.21 \times 10^{-7}$ $12$ $23$ sec $0.482819584$ $0.111335863$ $3.54 \times 10^{-5}$ $8$ $16$ sec
 $\lambda$ $\alpha$ $t_0$ cost iter cpu time $-10$ $0.990000000$ $0.337420581$ $1.33 \times 10^{-7}$ $14$ $14$ sec $0.386646216$ $0.337420581$ $0.001223$ $15$ $24$ sec $-9$ $0.990000000$ $0.331971040$ $5.73 \times 10^{-7}$ $13$ $16$ sec $0.388463867$ $0.331971040$ $0.000473$ $11$ $20$ sec $-8$ $0.990000000$ $0.325160189$ $5.07 \times 10^{-7}$ $13$ $14$ sec $0.390736229$ $0.325160189$ $0.000582$ $10$ $17$ sec $-7$ $0.990000000$ $0.316405255$ $3.36 \times 10^{-7}$ $12$ $18$ sec $0.393657966$ $0.316405255$ $0.000745$ $10$ $16$ sec $-6$ $0.990000000$ $0.304735585$ $3.79 \times 10^{-7}$ $11$ $14$ sec $0.397552887$ $0.304735585$ $9.84 \times 10^{-5}$ $12$ $20$ sec $-5$ $0.990000000$ $0.288405794$ $3.33 \times 10^{-7}$ $11$ $13$ sec $0.403001262$ $0.288405794$ $4.38 \times 10^{-5}$ $13$ $18$ sec $-4$ $0.990000000$ $0.263931464$ $2.09 \times 10^{-7}$ $11$ $14$ sec $0.411150343$ $0.263931464$ $2.25 \times 10^{-5}$ $9$ $16$ sec $-3$ $0.990000000$ $0.223215311$ $1.45 \times 10^{-7}$ $11$ $13$ sec $0.424571657$ $0.223215311$ $3.88 \times 10^{-5}$ $7$ $11$ sec $-2$ $0.990000000$ $0.142355886$ $1.70 \times 10^{-7}$ $10$ $17$ sec $0.448686986$ $0.142355886$ $1.30 \times 10^{-5}$ $7$ $13$ sec $-1$ $0.990000000$ $0.111335863$ $9.21 \times 10^{-7}$ $12$ $23$ sec $0.482819584$ $0.111335863$ $3.54 \times 10^{-5}$ $8$ $16$ sec
Means and standard deviations (std) of the number of iterations, computing time and computing time per iteration of the computational results shown in Table 9
 formulation iteration cpu time $\frac{\rm cpu\ time}{\rm iteration}$ mean std mean std mean std proposed $\approx 12 (11.8)$ $\approx 1 (1.23)$ $16.5$ $3.41$ $1.46$ $0.34$ classical $\approx 10 (10.2)$ $\approx 3 (2.61)$ $17.1$ $3.70$ $1.77$ $0.17$
 formulation iteration cpu time $\frac{\rm cpu\ time}{\rm iteration}$ mean std mean std mean std proposed $\approx 12 (11.8)$ $\approx 1 (1.23)$ $16.5$ $3.41$ $1.46$ $0.34$ classical $\approx 10 (10.2)$ $\approx 3 (2.61)$ $17.1$ $3.70$ $1.77$ $0.17$
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