doi: 10.3934/mcrf.2021049
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Continuity with respect to the speed for optimal ship forms based on Michell's formula

Laboratoire de Mathématiques et Applications, Université de Poitiers, CNRS, F-86073 Poitiers, France

* Corresponding author: M. Pierre

Received  June 2020 Revised  July 2021 Early access October 2021

Fund Project: This works benefited from the support of the project OFHYS of the CNRS 80|prime 2019 initiative.

We consider a ship hull design problem based on Michell's wave resistance. The half hull is represented by a nonnegative function and we seek the function whose domain of definition has a given area and which minimizes the total resistance for a given speed and a given volume. We show that the optimal hull depends only on two parameters without dimension, the viscous drag coefficient and the Froude number of the area of the support. We prove that, up to uniqueness, the optimal hull depends continuously on these two parameters. Moreover, the contribution of Michell's wave resistance vanishes as either the Froude number or the drag coefficient goes to infinity. Numerical simulations confirm the theoretical results for large Froude numbers. For Froude numbers typically smaller than $ 1 $, the famous bulbous bow is numerically recovered. For intermediate Froude numbers, a "sinking" phenomenon occurs. It can be related to the nonexistence of a minimizer.

Citation: Julien Dambrine, Morgan Pierre. Continuity with respect to the speed for optimal ship forms based on Michell's formula. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021049
References:
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G. Allaire, Conception Optimale De Structures, vol. 58 of Mathématiques & Applications, Springer-Verlag, Berlin, 2007.  Google Scholar

[2]

G. Allaire, C. Dapogny and F. Jouve, Shape and topology optimization, Geometric Partial Differential Equations. Part II, Handb. Numer. Anal., Elsevier/North-Holland, Amsterdam, 22 2021, 1–132. doi: 10.1016/bs.hna.2020.10.004.  Google Scholar

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G. Allaire and O. Pantz, Structural optimization with FreeFem++, Struct. Multidiscip. Optim., 32 (2006), 173-181.  doi: 10.1007/s00158-006-0017-y.  Google Scholar

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G. P. BenhamJ. P. BoucherR. LabbéM. Benzaquen and C. Clanet, Wave drag on asymmetric bodies, J. Fluid Mech., 878 (2019), 147-168.  doi: 10.1017/jfm.2019.638.  Google Scholar

[5]

L. Birk, Fundamentals of Ship Hydrodynamics: Fluid Mechanics, Ship Resistance and Propulsion, Wiley, 2019. doi: 10.1002/9781119191575.  Google Scholar

[6]

J.-P. BoucherR. LabbéC. Clanet and M. Benzaquen, Thin or bulky: Optimal aspect ratios for ship hulls, Phys. Rev. Fluids, 3 (2018), 074802.  doi: 10.1103/PhysRevFluids.3.074802.  Google Scholar

[7] A. Braides, $\Gamma$-Convergence for Beginners, vol. 22 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2002.  doi: 10.1093/acprof:oso/9780198507840.001.0001.  Google Scholar
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L. BrascoG. De Philippis and B. Velichkov, Faber-Krahn inequalities in sharp quantitative form, Duke Math. J., 164 (2015), 1777-1831.  doi: 10.1215/00127094-3120167.  Google Scholar

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H. Brezis, Analyse Fonctionnelle, Collection Mathématiques Appliquées Pour la Maîtrise, Masson, Paris, 1983.  Google Scholar

[10]

D. Bucur, Existence results, In Shape Optimization and Spectral Theory, De Gruyter Open, Warsaw, (2017), 13–28. doi: 10.1515/9783110550887-002.  Google Scholar

[11]

D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, Progress in Nonlinear Differential Equations and Their Applications, 65, Birkhäuser Boston, Inc., Boston, MA, 2005.  Google Scholar

[12]

M. Burger, A framework for the construction of level set methods for shape optimization and reconstruction, Interfaces Free Bound., 5 (2003), 301-329.  doi: 10.4171/IFB/81.  Google Scholar

[13]

G. Buttazzo, A survey on the Newton problem of optimal profiles, In Variational Analysis and Aerospace Engineering, Springer Optim. Appl., Springer, New York, 33 (2009), 33–48. doi: 10.1007/978-0-387-95857-6_3.  Google Scholar

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G. Buttazzo and B. Kawohl, On Newton's problem of minimal resistance, Math. Intelligencer, 15 (1993), 7-12.  doi: 10.1007/BF03024318.  Google Scholar

[15]

S. M. CalisalO. Goren and D. B. Danisman, Resistance reduction by increased beam for displacement type ships, J. Ship. Res., 46 (2002), 208-213.  doi: 10.5957/jsr.2002.46.3.208.  Google Scholar

[16]

J. DambrineE. Noviani and M. Pierre, Rankine-type cylinders having zero wave resistance in infinitely deep flows, IMA J. Appl. Math., 85 (2020), 343-364.  doi: 10.1093/imamat/hxaa008.  Google Scholar

[17]

J. Dambrine and M. Pierre, Regularity of optimal ship forms based on Michell's wave resistance, Appl. Math. Optim., 82 (2020), 23-62.  doi: 10.1007/s00245-018-9490-0.  Google Scholar

[18]

J. DambrineM. Pierre and G. Rousseaux, A theoretical and numerical determination of optimal ship forms based on Michell's wave resistance, ESAIM Control Optim. Calc. Var., 22 (2016), 88-111.  doi: 10.1051/cocv/2014067.  Google Scholar

[19]

C. DapognyP. FreyF. Omnès and Y. Privat, Geometrical shape optimization in fluid mechanics using FreeFem++, Struct. Multidiscip. Optim., 58 (2018), 2761-2788.  doi: 10.1007/s00158-018-2023-2.  Google Scholar

[20]

L. D. Ferreiro, The social history of the bulbous bow, Technology and Culture, 52 (2011), 335-359.  doi: 10.1353/tech.2011.0055.  Google Scholar

[21]

F. de Gournay, Velocity extension for the level-set method and multiple eigenvalues in shape optimization, SIAM J. Control Optim., 45 (2006), 343-367.  doi: 10.1137/050624108.  Google Scholar

[22]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[23]

O. GorenS. Calisal and D. B. Danisman, Mathematical programming basis for ship resistance reduction through the optimization of design waterline, J. Mar. Sci. Technol., 22 (2017), 772-783.  doi: 10.1007/s00773-017-0447-9.  Google Scholar

[24]

A. S. Gotman, The comperative criterion in deciding on the ship hull form with least wave resistance, In Proceedings Colloquium EUROMECH 374, (1998), 277–284. Google Scholar

[25]

A. S. Gotman, Study of Michell's integral and influence of viscosity and ship hull form on wave resistance, Oceanic Engineering International, 6 (2002), 74-115.   Google Scholar

[26]

A. S. Gotman, Navigating the wake of past efforts, Journal of Ocean Technology, 2 (2007), 74-96.   Google Scholar

[27]

F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.  doi: 10.1515/jnum-2012-0013.  Google Scholar

[28]

A. Henrot and M. Pierre, About critical points of the energy in an electromagnetic shaping problem, In Boundary Control and Boundary Variation (Sophia-Antipolis, 1990), Lect. Notes Control Inf. Sci., Springer, Berlin, 178 (1992), 238–252. doi: 10.1007/BFb0006699.  Google Scholar

[29]

A. Henrot and M. Pierre, Variation et Optimisation de Formes, vol. 48 of Mathématiques & Applications, Springer, Berlin, 2005. doi: 10.1007/3-540-37689-5.  Google Scholar

[30]

C.-C. Hsiung, Optimal ship forms for minimum wave resistance, J. Ship Res., 25 (1981), 95-116.  doi: 10.5957/jsr.1981.25.2.95.  Google Scholar

[31]

C.-C. Hsiung and S. Dong, Optimal ship forms for minimum total resistance, J. Ship Res., 28 (1984), 163-172.  doi: 10.5957/jsr.1984.28.3.163.  Google Scholar

[32]

A. A. Kostyukov, Theory of Ship Waves and Wave Resistance, Effective Communications Inc., Iowa City, Iowa, 1968. Google Scholar

[33]

M. G. Krein and V. G. Sizov, On the form of a ship of minimum total resistance (in Russian), Unpublished, 1960 Google Scholar

[34]

J. Lamboley and M. Pierre, Regularity of optimal spectral domains, In Shape Optimization and Spectral Theory, De Gruyter Open, Warsaw, (2017), 29–77. doi: 10.1515/9783110550887-003.  Google Scholar

[35]

Z. Lian-en, Optimal ship forms for minimal total resistance in shallow water, Schriftenreihe Schiffbau, 445 (1984), 1-60.   Google Scholar

[36]

W. C. Lin, W. C. Webster and J. V. Wehausen, Ships of Minimum Total Resistance, Technical Report No. NA-63-7, Institute of Engineering Research, University of California at Berkeley, 1963. Google Scholar

[37] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.   Google Scholar
[38]

J. P. Michalski, A. Pramila and S. Virtanen, Creation of ship body form with minimum theoretical resistance using finite element method, In Numerical Techniques for Engineering Analysis and Design, Springer Netherlands, (1987), 263–270. doi: 10.1007/978-94-009-3653-9_30.  Google Scholar

[39]

J. H. Michell, The wave resistance of a ship, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 45 (1898), 106-123.  doi: 10.1080/14786449808621111.  Google Scholar

[40]

F. C. Michelsen, Wave Resistance Solution of Michell's Integral for Polynomial Ship Forms, PhD thesis, University of Michigan, 1960. Google Scholar

[41]

F. Murat and J. Simon, Sur le contrôle par un domaine géométrique, Publication du Laboratoire d'Analyse Numérique de l'Université Paris VI, 222 pp. Google Scholar

[42]

V. G. Sizov, The seminar on ship hydrodynamics, organized by Professor M. G. Krein, In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Oper. Theory Adv. Appl., Birkhäuser, Basel, 117 (2000), 9–20.  Google Scholar

[43]

J. Sokoƚowski and J.-P. Zolésio, Introduction to Shape Optimization, Springer Series in Computational Mathematics, 16, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58106-9.  Google Scholar

[44] R. Temam and A. Miranville, Mathematical Modeling in Continuum Mechanics, 2$^nd$ edition, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9780511755422.  Google Scholar
[45]

E. Tuck and L. Lazauskas, Drag on a ship and Michell's integral, In Proceedings of the XXII International Congress of Theoretical and Applied Mechanics, Adelaide, Australia, 2008. Google Scholar

[46]

E. O. Tuck, The wave resistance formula of J. H. Michell (1898) and its significance to recent research in ship hydrodynamics, J. Austral. Math. Soc. Ser. B, 30 (1989), 365-377.  doi: 10.1017/S0334270000006329.  Google Scholar

[47]

E. O. Tuck, L. Lazauskas and D. C. Scullen, Sea Wave Pattern Evaluation - Part 1 Report: Primary Code and Test Results (surface vessels), Technical Report, Applied Mathematics Department, The University of Adelaide, Australia, 1999. Google Scholar

[48]

E. O. Tuck, D. C. Scullen and L. Lazauskas, Ship-wave patterns in the spirit of Michell, In IUTAM Symposium on Free Surface Flows, Fluid Mechanics and Its Applications, (eds. A. C. King and Y. D. Shikhmurzaev), Springer, Dordrecht, 62 (2001), 311–318. doi: 10.1007/978-94-010-0796-2_38.  Google Scholar

[49]

J. V. Wehausen, The wave resistance of ships, Advances in Applied Mechanics, Elsevier, 13 (1973), 93-245.  doi: 10.1016/S0065-2156(08)70144-3.  Google Scholar

show all references

References:
[1]

G. Allaire, Conception Optimale De Structures, vol. 58 of Mathématiques & Applications, Springer-Verlag, Berlin, 2007.  Google Scholar

[2]

G. Allaire, C. Dapogny and F. Jouve, Shape and topology optimization, Geometric Partial Differential Equations. Part II, Handb. Numer. Anal., Elsevier/North-Holland, Amsterdam, 22 2021, 1–132. doi: 10.1016/bs.hna.2020.10.004.  Google Scholar

[3]

G. Allaire and O. Pantz, Structural optimization with FreeFem++, Struct. Multidiscip. Optim., 32 (2006), 173-181.  doi: 10.1007/s00158-006-0017-y.  Google Scholar

[4]

G. P. BenhamJ. P. BoucherR. LabbéM. Benzaquen and C. Clanet, Wave drag on asymmetric bodies, J. Fluid Mech., 878 (2019), 147-168.  doi: 10.1017/jfm.2019.638.  Google Scholar

[5]

L. Birk, Fundamentals of Ship Hydrodynamics: Fluid Mechanics, Ship Resistance and Propulsion, Wiley, 2019. doi: 10.1002/9781119191575.  Google Scholar

[6]

J.-P. BoucherR. LabbéC. Clanet and M. Benzaquen, Thin or bulky: Optimal aspect ratios for ship hulls, Phys. Rev. Fluids, 3 (2018), 074802.  doi: 10.1103/PhysRevFluids.3.074802.  Google Scholar

[7] A. Braides, $\Gamma$-Convergence for Beginners, vol. 22 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2002.  doi: 10.1093/acprof:oso/9780198507840.001.0001.  Google Scholar
[8]

L. BrascoG. De Philippis and B. Velichkov, Faber-Krahn inequalities in sharp quantitative form, Duke Math. J., 164 (2015), 1777-1831.  doi: 10.1215/00127094-3120167.  Google Scholar

[9]

H. Brezis, Analyse Fonctionnelle, Collection Mathématiques Appliquées Pour la Maîtrise, Masson, Paris, 1983.  Google Scholar

[10]

D. Bucur, Existence results, In Shape Optimization and Spectral Theory, De Gruyter Open, Warsaw, (2017), 13–28. doi: 10.1515/9783110550887-002.  Google Scholar

[11]

D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, Progress in Nonlinear Differential Equations and Their Applications, 65, Birkhäuser Boston, Inc., Boston, MA, 2005.  Google Scholar

[12]

M. Burger, A framework for the construction of level set methods for shape optimization and reconstruction, Interfaces Free Bound., 5 (2003), 301-329.  doi: 10.4171/IFB/81.  Google Scholar

[13]

G. Buttazzo, A survey on the Newton problem of optimal profiles, In Variational Analysis and Aerospace Engineering, Springer Optim. Appl., Springer, New York, 33 (2009), 33–48. doi: 10.1007/978-0-387-95857-6_3.  Google Scholar

[14]

G. Buttazzo and B. Kawohl, On Newton's problem of minimal resistance, Math. Intelligencer, 15 (1993), 7-12.  doi: 10.1007/BF03024318.  Google Scholar

[15]

S. M. CalisalO. Goren and D. B. Danisman, Resistance reduction by increased beam for displacement type ships, J. Ship. Res., 46 (2002), 208-213.  doi: 10.5957/jsr.2002.46.3.208.  Google Scholar

[16]

J. DambrineE. Noviani and M. Pierre, Rankine-type cylinders having zero wave resistance in infinitely deep flows, IMA J. Appl. Math., 85 (2020), 343-364.  doi: 10.1093/imamat/hxaa008.  Google Scholar

[17]

J. Dambrine and M. Pierre, Regularity of optimal ship forms based on Michell's wave resistance, Appl. Math. Optim., 82 (2020), 23-62.  doi: 10.1007/s00245-018-9490-0.  Google Scholar

[18]

J. DambrineM. Pierre and G. Rousseaux, A theoretical and numerical determination of optimal ship forms based on Michell's wave resistance, ESAIM Control Optim. Calc. Var., 22 (2016), 88-111.  doi: 10.1051/cocv/2014067.  Google Scholar

[19]

C. DapognyP. FreyF. Omnès and Y. Privat, Geometrical shape optimization in fluid mechanics using FreeFem++, Struct. Multidiscip. Optim., 58 (2018), 2761-2788.  doi: 10.1007/s00158-018-2023-2.  Google Scholar

[20]

L. D. Ferreiro, The social history of the bulbous bow, Technology and Culture, 52 (2011), 335-359.  doi: 10.1353/tech.2011.0055.  Google Scholar

[21]

F. de Gournay, Velocity extension for the level-set method and multiple eigenvalues in shape optimization, SIAM J. Control Optim., 45 (2006), 343-367.  doi: 10.1137/050624108.  Google Scholar

[22]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[23]

O. GorenS. Calisal and D. B. Danisman, Mathematical programming basis for ship resistance reduction through the optimization of design waterline, J. Mar. Sci. Technol., 22 (2017), 772-783.  doi: 10.1007/s00773-017-0447-9.  Google Scholar

[24]

A. S. Gotman, The comperative criterion in deciding on the ship hull form with least wave resistance, In Proceedings Colloquium EUROMECH 374, (1998), 277–284. Google Scholar

[25]

A. S. Gotman, Study of Michell's integral and influence of viscosity and ship hull form on wave resistance, Oceanic Engineering International, 6 (2002), 74-115.   Google Scholar

[26]

A. S. Gotman, Navigating the wake of past efforts, Journal of Ocean Technology, 2 (2007), 74-96.   Google Scholar

[27]

F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.  doi: 10.1515/jnum-2012-0013.  Google Scholar

[28]

A. Henrot and M. Pierre, About critical points of the energy in an electromagnetic shaping problem, In Boundary Control and Boundary Variation (Sophia-Antipolis, 1990), Lect. Notes Control Inf. Sci., Springer, Berlin, 178 (1992), 238–252. doi: 10.1007/BFb0006699.  Google Scholar

[29]

A. Henrot and M. Pierre, Variation et Optimisation de Formes, vol. 48 of Mathématiques & Applications, Springer, Berlin, 2005. doi: 10.1007/3-540-37689-5.  Google Scholar

[30]

C.-C. Hsiung, Optimal ship forms for minimum wave resistance, J. Ship Res., 25 (1981), 95-116.  doi: 10.5957/jsr.1981.25.2.95.  Google Scholar

[31]

C.-C. Hsiung and S. Dong, Optimal ship forms for minimum total resistance, J. Ship Res., 28 (1984), 163-172.  doi: 10.5957/jsr.1984.28.3.163.  Google Scholar

[32]

A. A. Kostyukov, Theory of Ship Waves and Wave Resistance, Effective Communications Inc., Iowa City, Iowa, 1968. Google Scholar

[33]

M. G. Krein and V. G. Sizov, On the form of a ship of minimum total resistance (in Russian), Unpublished, 1960 Google Scholar

[34]

J. Lamboley and M. Pierre, Regularity of optimal spectral domains, In Shape Optimization and Spectral Theory, De Gruyter Open, Warsaw, (2017), 29–77. doi: 10.1515/9783110550887-003.  Google Scholar

[35]

Z. Lian-en, Optimal ship forms for minimal total resistance in shallow water, Schriftenreihe Schiffbau, 445 (1984), 1-60.   Google Scholar

[36]

W. C. Lin, W. C. Webster and J. V. Wehausen, Ships of Minimum Total Resistance, Technical Report No. NA-63-7, Institute of Engineering Research, University of California at Berkeley, 1963. Google Scholar

[37] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.   Google Scholar
[38]

J. P. Michalski, A. Pramila and S. Virtanen, Creation of ship body form with minimum theoretical resistance using finite element method, In Numerical Techniques for Engineering Analysis and Design, Springer Netherlands, (1987), 263–270. doi: 10.1007/978-94-009-3653-9_30.  Google Scholar

[39]

J. H. Michell, The wave resistance of a ship, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 45 (1898), 106-123.  doi: 10.1080/14786449808621111.  Google Scholar

[40]

F. C. Michelsen, Wave Resistance Solution of Michell's Integral for Polynomial Ship Forms, PhD thesis, University of Michigan, 1960. Google Scholar

[41]

F. Murat and J. Simon, Sur le contrôle par un domaine géométrique, Publication du Laboratoire d'Analyse Numérique de l'Université Paris VI, 222 pp. Google Scholar

[42]

V. G. Sizov, The seminar on ship hydrodynamics, organized by Professor M. G. Krein, In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Oper. Theory Adv. Appl., Birkhäuser, Basel, 117 (2000), 9–20.  Google Scholar

[43]

J. Sokoƚowski and J.-P. Zolésio, Introduction to Shape Optimization, Springer Series in Computational Mathematics, 16, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58106-9.  Google Scholar

[44] R. Temam and A. Miranville, Mathematical Modeling in Continuum Mechanics, 2$^nd$ edition, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9780511755422.  Google Scholar
[45]

E. Tuck and L. Lazauskas, Drag on a ship and Michell's integral, In Proceedings of the XXII International Congress of Theoretical and Applied Mechanics, Adelaide, Australia, 2008. Google Scholar

[46]

E. O. Tuck, The wave resistance formula of J. H. Michell (1898) and its significance to recent research in ship hydrodynamics, J. Austral. Math. Soc. Ser. B, 30 (1989), 365-377.  doi: 10.1017/S0334270000006329.  Google Scholar

[47]

E. O. Tuck, L. Lazauskas and D. C. Scullen, Sea Wave Pattern Evaluation - Part 1 Report: Primary Code and Test Results (surface vessels), Technical Report, Applied Mathematics Department, The University of Adelaide, Australia, 1999. Google Scholar

[48]

E. O. Tuck, D. C. Scullen and L. Lazauskas, Ship-wave patterns in the spirit of Michell, In IUTAM Symposium on Free Surface Flows, Fluid Mechanics and Its Applications, (eds. A. C. King and Y. D. Shikhmurzaev), Springer, Dordrecht, 62 (2001), 311–318. doi: 10.1007/978-94-010-0796-2_38.  Google Scholar

[49]

J. V. Wehausen, The wave resistance of ships, Advances in Applied Mechanics, Elsevier, 13 (1973), 93-245.  doi: 10.1016/S0065-2156(08)70144-3.  Google Scholar

Figure 1.  The domain of definition $ \omega $ of the hull $ f $
Figure 2.  Symmetrization $z\mapsto -z$
Figure 13.  $ J^\star $ vs $ Fr $ ($ C_F = 0.01 $)
Figure 14.  $ J^\star $ vs $ Fr $ for three different drag coefficients
Figure 3.  Wigley hull with length $ L = 2 $, draft $ T = 0.5 $ and beam $ B = 0.4 $
Figure 4.  The minimizing sequence of Wigley hulls $ w_{L,T_L,6} $ ($ L = 3 $, $ 4 $ and $ 5 $)
Figure 5.  $ J_{wave}(w_{L,T_L,6}) $ vs. $ L $
Figure 9.  A minimizing sequence for $ Fr = 1.75 $ (sinking case)
Figure 6.  Optimal domain for $ Fr = 0.46 $
Figure 7.  Optimal domains for $ Fr = 0.67 $ (top), $ 0.81 $ (middle) and $ 0.98 $ (bottom)
Figure 8.  Optimal hull for $ Fr = 0.67 $
Figure 10.  Optimal hull built on a half disk for $ Fr = 1.75 $ (3D view of Figure \ref{Fr1p75}-top)
Figure 11.  Optimal domains for $ Fr = 2.45 $ (top), $ 3.15 $ (middle) and $ 4.90 $ (bottom)
Figure 12.  Optimal hull for $ Fr = 4.90 $
Table 1.  Comparison of the area Froude number $ Fr $, the length $ L $ of the optimal domain and the length Froude number $ Fr_L $ ($ a = 1 $)
$ Fr $ 0.34 0.46 0.58 0.67 0.74 0.81 0.89 0.98
$ \sqrt{\pi}L $ 2.82 3.69 4.73 5.12 5.81 6.19 6.43 6.50
$ Fr_L $ 0.27 0.32 0.35 0.39 0.41 0.43 0.47 0.51
$ Fr $ 0.34 0.46 0.58 0.67 0.74 0.81 0.89 0.98
$ \sqrt{\pi}L $ 2.82 3.69 4.73 5.12 5.81 6.19 6.43 6.50
$ Fr_L $ 0.27 0.32 0.35 0.39 0.41 0.43 0.47 0.51
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