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Naval Surface Warfare Center Carderock Division West Bethesda, MD 20817-5700

NSWCCD-50-TR–2003/33 August 2003

Hydrodynamics Directorate Technical Report

Shape Optimization in Free Surface Potential Flow Using an Adjoint Formulation by Saad A. Ragab

Approved for public release; distribution is unlimited.

Naval Surface Warfare Center Carderock Division West Bethesda, MD 20817-5700

Approved for public release; distribution is unlimited.

NSWCCD-50-TR–2003/33 August 2003 Hydrodynamics Directorate

Technical Report

Shape Optimization in Free Surface Potential FlowUsing an Adjoint Formulation

by Saad A. Ragab

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4. TITLE AND SUBTITLE

Shape Optimization in Free Surface Potential Flow Using an Adjoint Formulation

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6. AUTHOR(S)

Saad A. Ragab

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Naval Surface Warfare Center Carderock Division (Code 50) 9500 Macarthur Boulevard West Bethesda, MD 20817-5700

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12. DISTRIBUTION / AVAILABILITY STATEMENT Approved for public release; distribution is unlimited.

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14. ABSTRACT In this report, a numerical method for shape optimization of surface ships and submarines operating near a free surface is presented. The classical potential flow theory is used, and the free surface boundary conditions are linearized using Kelvin’s or Dawson’s method. Several objective functionals are shown for (a) wave resistance minimization and (b) inverse problems where a target pressure distribution on hull or a free surface wave pattern is prescribed. An important contribution of this work is the formulation of an adjoint approach for computing the gradients of these objective functionals. The potential flow problem is solved using a panel code (SWAN-v2.2). Like the velocity potential function, the adjoint function is governed by Laplace’s equation, however, the adjoint radiation (uniqueness) condition demands that waves may exist only upstream. The adjoint problem is also solved using the same code (SWAN-v2.2) after some modifications are introduced to handle the respective boundary conditions. 15. SUBJECT TERMS Optimization; adjoint; potential flow; free surface

16. SECURITY CLASSIFICATION OF: 19a. NAME OF RESPONSIBLE PERSON Arthur M. Reed

a. REPORT UNCLASSIFIED

b. ABSTRACT UNCLASSIFIED

c. THIS PAGE UNCLASSIFIED

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122

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Standard Form 298 (Rev. 8-98)

Prescribed by ANSI Std. Z39.18i

(BLOCK 19) The hull geometry is parameterized by B-spline surface patches whose control-points offsets are used as design variables. The optimized hull is constrained to have equal displacement as a base-line hull, but its wetted surface is not allowed to increase above a given margin. To prevent bizarre shapes, lower and upper bounds are also specified on control points. This constrained optimization problem is solved by a generalized reduced gradient method. The method has been applied successfully to several examples of submerged bodies and surface ships. A base-line hull is first defined. At a given Froude number, the hull is optimized for minimum wave resistance or for a prescribed pressure distribution on hull. The target pressure used in this study is the double-body pressure distribution on the base-line hull. This is the pressure that would exist on the base-line hull if the free surface is replaced by a rigid lid; also known as zero-Froude-number flow. Other pressure distributions can be easily specified. The accuracy and efficiency of the adjoint approach are demonstrated by comparisons with direct calculations of the gradients using a finite-difference method. Results also include geometric characteristics, wave resistance, and surface wave patterns of optimized hull forms.

ii

NSWCCD-50-TR-2003/33

Contents

Page

Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Administrative Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Wave Resistance Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Gradient-Based Optimization Techniques . . . . . . . . . . . . . . . . . . . . . . . 6

An Adjoint Formulation For Free-Surface Flow . . . . . . . . . . . . . . . . . . 8

The Potential Flow Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Objective Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

The Adjoint Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

The Gradient dL/dθ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Dawson’s Linearized Free-Surface Condition . . . . . . . . . . . . . . . . . . . . . 24

Submarines Operating Near a Free Surface . . . . . . . . . . . . . . . . . . . . 26

Geometry Parameterization by B-Spline Surfaces . . . . . . . . . . . . . . . . . . 26

Base-Line Submarine G0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Wave Resistance Minimization at Froude Number Fn = 0.230 . . . . . . . . . . . 27


Using the Depths of Control Points as Design Variables . . . . . . . . . . . . . . . 30

Wave Resistance Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Design for a Target Pressure Distribution . . . . . . . . . . . . . . . . . . . . 31

Surface Ships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Base-Line Ship G0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59


Optimization Using Finite Difference for Gradient Computation . . . . . . . . . . 61

A Target Pressure Distribution at Froude Number Fn = 0.287 . . . . . . . . . . . 62

Weakly Wall-Sided Design at Froude Number Fn = 0.287 . . . . . . . . . . . . . . 64

Partially Constrained Stern at Froude Number Fn = 0.287 . . . . . . . . . . . . . 64

Conclusions and Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . 105

Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . 106

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

iii


Figures

Page

1. A submarine model with a sail and its wave pattern at Froude number Fn =0.192 and axis depth = 0.12, (Reference length = body length). . . . . . . . . 3

2. A submarine model with two sails and its wave pattern at Froude numberFn = 0.192 and axis depth = 0.12, (Reference length = body length). . . . . . 4

3. A sketch of flow domain boundaries: Free Surface (FS), Body Surface (BS), andFar Field Surface (FFS). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4. A sketch of free surface boundaries and waterline. . . . . . . . . . . . . . . . . 18

5. Indices of bi-cubic B-spline control points for a submarine configuration. . . . 35

6. Base-line submarine G0, z = 0 is the free surface. . . . . . . . . . . . . . . . . 36

7. Wave resistance coefficient for base-line submarine G0 (Rp by pressure integra-tion and Rw by wave cut method). . . . . . . . . . . . . . . . . . . . . . . . . 36

8. Gradient computed by adjoint and finite-difference methods, Fn = 0.230. . . . 37

9. Errors in gradient components normalized by gradient vector magnitude, Fn =0.230. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

10. A view of designed hull Gd2 sub230. . . . . . . . . . . . . . . . . . . . . . . . . 38

11. Projection on the xz-plane of designed hull Gd2 sub230. . . . . . . . . . . . . . 39

12. Projection on the xy-plane of designed hull Gd2 sub230. . . . . . . . . . . . . . 39

13. Cross sections of designed hull Gd2 sub230. . . . . . . . . . . . . . . . . . . . . 40

14. Wave pattern of base-line hull G0 (lower half) and designed hull Gd2 sub230(upper half), Fn = 0.230. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

15. Wave elevation in plane of symmetry of base-line hull G0 and designed hull Gd2

sub230, Fn = 0.230. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

16. Pressure contours (0.5F 2nCp) on top of base-line hull G0, Fn = 0.230. . . . . . . 43

17. Pressure contours (0.5F 2nCp) on top of designed hull Gd20 sub230, Fn = 0.230. . 43

18. Pressure contours (0.5F 2nCp) on side of base-line hull G0 sub230, Fn = 0.230. . 44

19. Pressure contours (0.5F 2nCp) on side of designed hull Gd2 sub230, Fn = 0.230. . 44

20. Gradient computed by adjoint and finite-difference methods, Fn = 0.300. . . . 45

21. Errors in gradient components normalized by gradient vector magnitude, Fn =0.300. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

22. Reduction of cost functional with design cycles for Gd10 sub300. . . . . . . . . 46

23. A view of designed hull Gd10 sub300. . . . . . . . . . . . . . . . . . . . . . . . 47

24. Projection on the xz-plane of designed hull Gd10 sub300. . . . . . . . . . . . . 48

25. Projection on the xy-plane of designed hull Gd10 sub300. . . . . . . . . . . . . 48

26. Bow view of designed hull Gd10 sub300. . . . . . . . . . . . . . . . . . . . . . . 49

27. Stern view of designed hull Gd10 sub300. . . . . . . . . . . . . . . . . . . . . . 50

iv


Figures (Continued)

Page

28. Wave resistance coefficient for base-line hull G0 and designed hull Gd10 sub300 (Rp

by pressure integration and Rw by wave cut method). . . . . . . . . . . . . . . 51

29. Wave pattern of base-line hull G0 (lower half) and designed hull Gd10 sub300(upper half). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

30. Wave elevation in the plane of symmetry of base-line hull G0 and designed hullGd10 sub300. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

31. Pressure contours (0.5F 2nCp) on bow of base-line hull G0 (left half) and designed

hull Gd10 sub300 (right half). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

32. Pressure contours (0.5F 2nCp) on stern of base-line hull G0 (left half) and designed

hull Gd10 sub300 (right half). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

33. Contribution of station strips (between x and x+dx) to wave resistance of base-linehull G0 and designed hull Gd10 sub300. . . . . . . . . . . . . . . . . . . . . . . 56

34. Contribution of all strips between bow (x = 0.5) and station x to wave resistanceof base-line hull G0 and designed hull Gd10 sub300. . . . . . . . . . . . . . . . 56

35. Designed body Gd sub450 for optimal solution to the inverse problem; free surfacez = 0, Froude number Fn = 0.450. . . . . . . . . . . . . . . . . . . . . . . . . . 57

36. Pressure coefficient distributions on the top line of symmetry of designed body Gd

sub450 and initial body G0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

37. Indices of B-spline control points for base-line surface ship G0. . . . . . . . . . . 66

38. Base-line hull G0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

39. Projection on the xy-plane of base-line hull G0. . . . . . . . . . . . . . . . . . . . 68

40. Body plan of base-line hull G0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

41. Wave resistance coefficient for base-line hull G0 (Rp by pressure integration andRw by wave cut method). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

42. Gradient of cost function computed by finite-difference and adjoint methods, hullG0, Fn = 0.287. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

43. Errors in gradient components normalized by gradient vector magnitude, Fn = 0.287. 70

44. Gradient computed with and without contribution of adjoint (ψ = 0) and finite-difference method, Fn = 0.287. . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

45. Errors in gradient components normalized by gradient vector magnitude, Fn = 0.287. 71

46. Reduction of cost functional with design cycles for Gd20 S287. . . . . . . . . . . . 72

47. Water-line view of designed hull Gd20 S287. . . . . . . . . . . . . . . . . . . . . . 73

48. Keel view of designed hull Gd20 S287. . . . . . . . . . . . . . . . . . . . . . . . . 74

49. Projection on the xy-plane of designed hull Gd20 S287. . . . . . . . . . . . . . . . 75

50. Body plan of designed hull Gd20 S287. . . . . . . . . . . . . . . . . . . . . . . . . 75

v


Figures (Continued)

Page

51. Wave resistance coefficient for base-line hull G0 and designed hull Gd20 S287 (Rp


52. Wave pattern on base-line hull G0 (lower half) and designed hull Gd20 S287 (upperhalf). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

53. Water-line wave elevation on base-line hull G0 and designed hull Gd20 S287. . . . 78

54. Pressure contours (0.5F 2nCp) on bow of base-line hull G0 (left half) and designed

hull Gd20 S287 (right half). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

55. Pressure contours (0.5F 2nCp) on stern of base-line hull G0 (left half) and designed

hull (Gd20 S287 (right half). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

56. Contribution of station strips (between x and x+dx) to wave resistance of base-linehull G0 and designed hull Gd20 S287. . . . . . . . . . . . . . . . . . . . . . . . 81

57. Contribution of all strips between bow x = 0.5 and station x to wave resistanceof base-line hull G0 and designed hull Gd20 S287. . . . . . . . . . . . . . . . . . 82

58. Comparison of hull forms by adjoint (right half) and finite difference (left half),bow view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

59. Comparison of hull forms by adjoint (right half) and finite difference (left half),stern view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

60. Comparison of wave pattern on finite-difference hull (lower half) and adjoint hull(upper half). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

61. Comparison of water-line wave elevation on finite-difference hull and adjoint hull. 86

62. Contribution of all strips between bow x = 0.5 and station x to wave resistanceof finite-difference hull, adjoint hull, and base-line hull. . . . . . . . . . . . . . 87

63. Wave resistance coefficient for finite-difference hull and adjoint hull (Rw by wavecut method). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

64. Sectional area distribution for finite-difference hull and adjoint hull. . . . . . . . 89

65. Waterline for finite-difference hull and adjoint hull. . . . . . . . . . . . . . . . . . 90

66. Gradient computed by finite-difference and adjoint methods, hull G0, Fn = 0.287. 91

67. Error in gradient components normalized by magnitude of gradient vector, hullG0, Fn = 0.287. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

68. Reduction of cost functional for hull Gd10 S287P. . . . . . . . . . . . . . . . . . . 92

69. Pressure distribution on hull, P = pressure on designed hull Gd10 S287P, Pd =target pressure, and P0 = pressure on G0, Fn = 0.287. . . . . . . . . . . . . . . 93

70. Water-line view of designed hull Gd10 S287P. . . . . . . . . . . . . . . . . . . . . 94

71. Keel view of designed hull Gd10 S287P. . . . . . . . . . . . . . . . . . . . . . . . 95

72. Projection on the xy-plane of designed hull Gd10 S287P. . . . . . . . . . . . . . . 96

73. Body plan of designed hull Gd10 S287P. . . . . . . . . . . . . . . . . . . . . . . . 96

vi


Figures (Continued)

Page

74. Wave resistance coefficient for base-line hull G0 and designed hull Gd20 S287P (Rp


75. Projection on the xy-plane of designed hull Gd20 S287WS. . . . . . . . . . . . . . 98

76. Body plan of designed hull Gd20 S287WS. . . . . . . . . . . . . . . . . . . . . . . 98

77. Water-line wave elevation on hull Gd20 S287 and weakly wall-sided hull Gd20 S287WS. 99

78. Water-line view of designed hull Gd20 S287B. . . . . . . . . . . . . . . . . . . . . 100

79. Projection on the xy-plane of designed hull Gd20 S287B. . . . . . . . . . . . . . . 101

80. Body plan of designed hull Gd20 S287B. . . . . . . . . . . . . . . . . . . . . . . . 101

81. Comparison of wave pattern on hull Gd20 S287 (lower half) and partially con-strained stern hull Gd20 S287B (upper half). . . . . . . . . . . . . . . . . . . . 102

82. Water-line wave elevation on hull Gd20 S287 and partially constrained stern hullGd20 S287B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

83. Contribution of all strips between bow x = 0.5 and station x to wave resistance ofbase-line hull G0, designed hull Gd20 S287, and partially constrained stern hullGd20 S287B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Tables

Page

1. dF/dyk by adjoint and central finite difference, minimum wave resistance, andoffsets yk of B-spline control points . . . . . . . . . . . . . . . . . . . . . . . . 33

2. dF/dzk by adjoint and central finite difference, minimum wave resistance, anddepths zk of B-spline control points . . . . . . . . . . . . . . . . . . . . . . . . 33

3. dF/dyk by adjoint and central finite difference, prescribed surface pressure, andoffsets yk of B-spline control points . . . . . . . . . . . . . . . . . . . . . . . . 34

4. dF/dzk by adjoint and central finite difference, prescribed surface pressure, anddepths zk of B-spline control points . . . . . . . . . . . . . . . . . . . . . . . . 34

vii


Administrative Information

Final report on research conducted by the author while on sabbatical at the CarderockDivision Naval Surface Warfare Center (NSWCCD), West Bethesda, MD, under the Inter-governmental Personnel Act assignment to the Office of Naval Research, Arlington, VA,during the year August 10, 1999 to August 9, 2000.

Acknowledgments

I wish to thank Dr. L. Patrick Purtell for the opportunity to conduct this research as apart of the Intergovernmental Personnel Act (IPA) assignment to the Office of Naval Research(ONR) during the year August 10, 1999 to August 9, 2000. Special thanks are due to Dr.Spiro G. Lekoudis at ONR for encouragement and insightful discussions. I am also indebtedto Dr. Arthur M. Reed (NSWCCD Code 5050) for numerous helpful ideas and suggestionsregarding the interpretation of results. This work could not have been completed withoutthe help of Mr. John Telste (NSWCCD Code 5400); I thank him for providing technicalsupport with the SWAN-v2.2 computer code and for many helpful discussions.

I wish to thank Dr. William Smith (NSWCCD Code 5030) for the warm hospitalityand all the support during the IPA year. I also thank Drs. Dane Hendrix, Thomas Fu,and William Faller (NSWCCD Code 50) for their assistance with computer networking andprinting. I have also enjoyed the friendship of Mr. Michael Smith and Mr. Samuel Balboa(Sam); thanks for the good times and support.

It is a pleasure to acknowledge Virginia Polytechnic Institute and State University forsponsoring my study-research leave (sabbatical) during the academic year August 10, 1999to May 9, 2000. I wish to thank Mr. S. M. Shin, a Ph. D. candidate in the Aerospace andOcean Engineering Department, for helpful discussions on free surface flows.

viii


Summary

This report presents a numerical method for shape optimization of surface ships andsubmarines operating near a free surface. The classical potential flow theory is used, and thefree surface boundary conditions are linearized using Kelvin’s or Dawson’s method. Severalobjective functionals are shown for (a) wave resistance minimization and (b) inverse problemswhere a target pressure distribution on hull or a free surface wave pattern is prescribed.

The formulation of an adjoint approach for computing the gradients of these objectivefunctionals is an important contribution of this work. The potential flow problem is solvedusing a panel code (SWAN-v2.2). Like the velocity potential function, the adjoint functionis governed by Laplace’s equation, however, the adjoint radiation (uniqueness) conditiondemands that waves may exist only upstream. The adjoint problem is also solved using thesame code (SWAN-v2.2) after some modifications are introduced to handle the respectiveboundary conditions.

The hull geometry is parameterized by B-spline surface patches whose control-pointsoffsets are used as design variables. The optimized hull is constrained to have equal dis-placement as a base-line hull, but its wetted surface is not allowed to increase above a givenmargin. To prevent bizarre shapes, lower and upper bounds are also specified on controlpoints. This constrained optimization problem is solved by a generalized reduced gradientmethod.

The method has been applied successfully to several examples of submerged bodies andsurface ships. A base-line hull is first defined. At a given Froude number, the hull is optimizedfor minimum wave resistance or for a prescribed pressure distribution on hull. The targetpressure used in this study is the double-body pressure distribution on the base-line hull.This is the pressure that would exist on the base-line hull if the free surface is replacedby a rigid lid; also known as zero-Froude-number flow. Other pressure distributions canbe easily specified. The accuracy and efficiency of the adjoint approach are demonstratedby comparisons with direct calculations of the gradients using a finite-difference method.Results also include geometric characteristics, wave resistance, and surface wave patterns ofoptimized hull forms.

Introduction

Marine vehicles design can be improved significantly by using automatic shape optimiza-tion techniques. These techniques can be used to control flow separation, wave resistance,and cavitation on submarines, ships and hydrofoils. In this report, an efficient numericalmethod for shape optimization is presented. The salient feature of the method is the use ofan adjoint formulation for computing the gradient of a cost functional. The adjoint approachis very efficient when the number of design variables is large; it is practically independent ofthat number. This allows more flexibility in the design of complex surfaces. Two problemsare considered: (a) the design of hull forms of minimum wave resistance and (b) optimal

1


solution to the inverse problem in free-surface potential flow. Since panel methods are com-monly used in the early stages of design of marine vehicles, the method presented here alsouses a panel code for solving the field and adjoint equations. Such a fast design code will bea valuable tool in the hands of the hydrodynamicist.

Wave Resistance Minimization

The orderly waves created by a ship on the face of calm water are beautiful to watch,yet they are undesirable. A significant amount of energy must be spent to sustain theirgeneration. This energy is carried away and never recovered by the ship, and hence itrepresents wave resistance. Ship waves are detectable and may contain information not tobe revealed to adversaries. These waves can also cause erosion of rivers banks and canals.

One of the objectives of hydrodynamic design of hull forms is to eliminate or minimizewave resistance. Apparently, this objective has been pursued for more than a hundred yearssince Kelvin (1887) published his work on ship waves. The principle at work for achievingminimum wave resistance is destructive (favorable) wave interference. Kelvin (1905) ide-alized the ship as a two-dimensional moving surface pressure distribution and showed thatthe waves produced are almost sinusoidal. He also demonstrated that, by adjusting theseparation distance between two identical distributions, the waves interfere destructively toproduce a waveless ship. Froude (1877) extensively studied the wave pattern produced by aship in steady motion. He showed that the humps and dips in the variation of wave resis-tance with Froude number are manifestations of the constructive and destructive interferencebetween the bow and stern transverse wave systems. Wave interference is also the underly-ing mechanism for the success of bulbous bows in reducing wave resistance. However, theireffectiveness is limited to low Froude numbers. Based on a paper by Eckert and Sharma(1970), Wehausen (1973) reported “The unanticipated saving in wave resistance is a resultof improved flow near the bow that avoids loss of energy through wave breaking.”

In addition to many examples in the literature, an interesting example of waves cancel-lation by destructive interference is presented here. The wave pattern due to a submarinemodel, which is an ellipsoid (length/diameter ratio = L/D = 8) fitted with a sail, is shownin Figure 1. The axis depth is 0.12 L and Froude number based on length is 0.192. The sail’stop is 0.018L below the free surface. As shown in the figure, strong transverse waves domi-nate the wave pattern. They can be eliminated easily by installing a second sail downstreamof the existing one. The wave pattern due the same model but with two sails is shown inFigure 2 which clearly shows the cancellation of transverse waves at this Froude number.The wave drag per unit displacement for the model with one sail is 0.0642, and for the modelwith two sails is 0.0265.

The search for waveless ships continues in the work of Tuck (1991). His approach issomewhat different, he does not rely on interference between the wave systems of bow andstern. Instead, he attempts to design a waveless stern, and by reversing the flow, he alsoobtains a waveless bow. More recently, Tulin and Oshri (1996) have used the principle of wave

2


X

Y

-2.5 -2 -1.5 -1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

0.001-0.001

Figure 1. A submarine model with a sail and its wave pattern at Froude numberFn = 0.192 and axis depth = 0.12, (Reference length = body length).

3


X

Y

-2.5 -2 -1.5 -1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

0.001-0.001

Figure 2. A submarine model with two sails and its wave pattern at Froudenumber Fn = 0.192 and axis depth = 0.12, (Reference length = bodylength).

4


cancellation to synthesize wave-free compound singularities. By using their method, theymodified the Wigley hull so that its wave resistance is reduced by a factor of 0.3 at Froudenumber of 0.5, however, the practicality of the modified hull remains to be established.

A more direct and fruitful approach is to use optimization methods and search for thehull geometry that minimizes the wave resistance. Michell (1898) formulated and solved thethin-ship problem. He obtained the wave resistance as an integral that depends explicitly onthe hull geometry. Early works on finding optimum hull forms minimized Michell’s integral.Wehausen (1973) gives an excellent review and summary of important results on ships ofminimum resistance. Of the vast literature on the subject, we mention only two examples:Hsiung (1981) who used Michell’s integral and Wyatt and Chang (1994) who used slender-ship theory.

The shortcomings of thin-ship theory and the related slender-ship theory are well known.With the advent of modern computers, more accurate numerical treatments of the ship prob-lem are possible. The exact potential flow problem can be solved by panel methods. The hullsurface boundary condition and the nonlinear free-surface conditions can be satisfied exactly.The exact treatment of the nonlinear free-surface boundary conditions requires much morecomputer time than the linearized conditions. For optimization problems, it is essential tohave a fast flow solver. This is because of the many evaluations of the flow solution called forby the optimization technique. The Neumann-Kelvin and Neumann-Dawson formulations,in which the free-surface conditions are linearized but the hull surface condition is exactly en-forced, have been widely used for wave resistance calculations. Although inconsistent, theseformulations are acceptable for optimization of hull forms because of the efficiency offered bythe linear free-surface condition. Refinements of optimized hulls should be eventually basedon the exact nonlinear conditions and include viscous effects as well.

Few articles have been published on optimization of hull forms using numerical solution tothe ship problem. To the knowledge of this writer, Lowe et al.’s (1994) is the only publishedwork that uses this approach. They designed a yacht hull for minimum wave resistancesubject to certain constraints. The salient feature of their work is the use of the partialdifferential equation method (Bloor and Oleksiewicz 1995) to generate the hull surface. Thismethod enables the shape of the hull to be completely specified using only a small numberof parameters while not overly limiting the range of obtainable shapes. They computed thewave resistance by using a panel method similar to that developed by Dawson and evaluatedthe gradient of the cost functional by finite differences.

Inverse Problems

Traditionally, hydrodynamic design has been posed as an inverse problem in which thebody shape is unknown and the desired performance is expressed through a pressure dis-tribution. A well known example is due to Lighthill (1945) who solves the inverse problemfor a two-dimensional hydrofoil. An exact and feasible solution to an inverse problem maynot exist. Shape optimization techniques aim at finding the best shape that produces the

5


target pressure distribution. The inverse formulation is desirable when a clear idea about thepressure distribution is at hand, such as the case of flow cavitation and separation control orshock-free flow in the design of transonic wings. Some of the recent studies on inverse prob-lems include Pashin et al. (1996) who present a method for finding three-dimensional bodyforms from given surface pressure distribution. They demonstrate their method for a bodyof revolution at an angle of attack, and obtain blunt noses for best cavitation performance.Huang et al. (1998) use an optimization technique to solve the inverse problem for a surfaceship.

The presence of a free surface offers an alternative formulation of the inverse problem.It may be feasible to seek a hull form so that its wave pattern at a given Froude numbermatches a target pattern as nearly as possible. A target pattern may be provided by thedouble-body pressure distribution on the free surface (a rigid lid in this flow). The pressurecoefficient distribution, Cp, on the rigid lid is interpreted as a wave pattern η0 = 0.5F 2

nCp,where Fn is the Froude number. This pattern has a near-field disturbance but no waves inthe far field and hence can be used to reduce wave resistance and other undesirable waveeffects.

Gradient-Based Optimization Techniques

Gradient-based optimization techniques such as conjugate gradient and quasi-Newtonmethods (e. g. Vanderplaats 1998) are commonly used because of their efficiency. An impor-tant step in their implementation is an accurate and fast evaluation of the gradient of thecost functional. A finite-difference method can be used but it becomes very inefficient if thenumber of design variables is large because it requires solving the field equations as manytimes as the number of design variables just to obtain the gradient.

Alternatively, an adjoint approach (e. g. Jameson 1988, Pironneau 1984, and Soemarwoto1997) avoids this difficulty by treating the field equations as constraints on the variationsin flow variables (the velocity potential). The constrained problem is then solved usingLagrange multipliers which are defined so that the first variation of the Lagrangian withrespect to flow variables vanishes. The governing equations of the Lagrange multipliers (alsocalled adjoint variables or co-states) have to be solved only once on an unperturbed shape.The effort of solving these equations is comparable to that of the field equations. The restof the terms in the gradient still depend on the number of control parameters but the costof their calculations is much less than solving the field equations. Their calculations requireonly geometric characteristics of the perturbed shapes but no field equations are solved onthese shapes.

The adjoint approach is well known in the mathematical theory for the control of systemsgoverned by partial differential equations (Lions, 1971). Applications of this approach toproblems governed by elliptic equations (e. g. potential flow problems) have been presentedby Pironneau (1984). Recently, Jameson (1988, 1995) extended this approach to problemsgoverned by the compressible Euler and Navier-Stokes equations for the design of transonic

6


finite wings. We have applied Jameson’s method to the inverse problem of three-dimensionaldeeply submerged configuration (Ragab, 1997). The new contributions of this report are theformulation of the adjoint problem for free-surface flows and its implementation in a panelcode for the hydrodynamic design of surface ships as well as bodies submerged near a freesurface.

7


An Adjoint Formulation For Free-Surface Flow

The Potential Flow Problem

We consider a ship in a steady rectilinear motion with constant velocity in a calm sea.Figure 3 defines a coordinate system that is fixed in the ship. The x-axis is positive in thedirection of ship velocity, the z-axis is positive vertically upward and the y-axis completesa right-hand system. The plane z = 0 coincides with the undisturbed free surface, and theplane y = 0 is assumed to be a plane of symmetry for the ship and flow field. In this system,we assume the flow to be steady, inviscid and irrotational. We linearize the free-surfaceconditions using either Kelvin’s or Dawson’s method. In the first method, the basis flow,around which the conditions are linearized, is a uniform flow whereas in the second methodthe basis flow is the double-body flow. In this section we use Kelvin’s linearization. Themodifications due to Dawson’s linearization are presented at the end of this section on theadjoint formulation beginning on page 24. The no-penetration condition is exactly enforcedon the hull surface. To formulate the problem, we select a flow domain Ω that is boundedby the hull surface BS, the free surface FS and a far field surface FFS. The union of thethree surfaces is denoted by Σ = BS ∪ FS ∪ FFS. The body surface is parameterized by aset of geometric parameters (θi, i = 1, . . . , N); in the following we will use θ to denote anyof the members of the set.

In terms of the disturbance velocity potential φ, the flow field is governed by:

∇2φ = 0 in Ω, (1)

∇φ · n + U∞ · n = 0 on BS, (2)

∂2φ

∂x2+ κ

∂φ

∂z= 0 on FS, (3)

and

φ = 0 on FFS, (4)

where n is the unit normal to the ship hull; it is directed from the water into the inside ofhull, and κ = g0/U

2∞ where g0 is the gravitational acceleration. In Equation (2) U∞ = −U∞ı

is the fluid velocity far upstream of the ship and ı is the unit vector in the x-direction. Wenote that boundary condition Equation (4) is valid only if the flow domain is unbounded (noside walls or bottom) and the far field surface FFS is removed to infinite distance from theship.

A radiation (uniqueness) condition that allows no waves to propagate upstream mustalso be added to the above equations (Stoker, 1957). We write such a condition as

φ and∂φ

∂x→ 0 as x → +∞. (5)

8


y

z

x

FS

BS

FFS

U

Figure 3. A sketch of flow domain boundaries: Free Surface (FS), Body Surface(BS), and Far Field Surface (FFS).

9


Panel methods are very efficient for solving this problem. An existing code (SWAN-v2.2) isused to solve for the velocity potential and other flow variables.

Objective Functionals

Shape optimization starts with the selection of an objective (or cost) functional to beminimized with respect to the hull geometric parameters θ. A candidate objective functionalis the wave resistance:

F = −∫

BS

pnx dS, (6)

where p is the pressure and nx is the x-component of the unit normal. Another method forcomputing wave resistance follows from global conservation of linear momentum (Wehausen,1973), which is also the basis for the wave-cut method,

F = −1

2ρ

∫WS

[(∂φ

∂x

)2

−(

∂φ

∂y

)2

−(

∂φ

∂z

)2]

dS +1

2ρg0

∫ ∞

−∞η2

0 dy, (7)

where ρ is the fluid density, WS is a vertical plane in the far wake normal to the ship’svelocity and η0 is the wave elevation relative the undisturbed free surface.

A second choice for the objective functional is

F =

∫FS

(η0 − η0d)2 dS (8)

where η0d is a target wave pattern. A hull form that minimizes this integral will have a wavepattern very close to η0d.

A third choice for the objective functional is still possible. We seek a hull geometry whosesurface pressure distribution is as nearly as possible to a prescribed (target) pressure. In theleast-squares sense, the objective functional is given by

F =1

2

∫BS

(p − pd)2 dS, (9)

where pd is the target pressure distribution. The last two objective functionals are classifiedas inverse problems. In the absence of clear ideas about what pd and η0d should be, thedouble-body flow offers useful suggestions. The double-body flow over an existing hull isdetermined, and the pressure coefficient distributions on the hull and free surface—nowconsidered as a rigid lid—can be used to specify a target pressure pd or a wave pattern η0d.

In this report, an objective functional of the general form:

F =

∫BS

f(θ, φ) dS +

∫FS

g(θ, φ) dS (10)

10


is considered, where f and g are given functions of θ and φ. The function g should be definedso that the free-surface integral exists. The case in which f or g is a function of pressurewill be considered afterwards.

The Adjoint Problem

To minimize F with respect to the set θ using gradient-based optimization methodssuch as the steepest descent, conjugate gradient, or the quasi-Newton method, the gradientdF/dθ is needed. Because of the large number of control parameters, use of finite differencesfor evaluating the gradient is very expensive. The adjoint approach is an efficient methodfor finding the gradient at the expense of solving an additional boundary-value problemsimilar to the original problem for the velocity potential. This approach is widely used inaerodynamic design. An important contribution of the present work is the formulation ofthe adjoint approach for free-surface flow and its implementation in a panel code. In thiswork, modifications have been introduced into SWAN-v2.2 code to solve the adjoint problemformulated here.

We note that a small variation in θ produces a variation in F that can be decomposedinto three components. For example, if the function f represents the pressure at a surfaceelement on the hull, then perturbing the hull geometry produces new pressure distributionat the element which is now displaced to a new position in the new flow field. This impliestwo sources of variations in F . A third component results from the change in the measureof the surface element dS. We assume that the hull surface and the free surface each can bemapped into a rectangular domain (ξ, η) and express the surface element by

dS = H(ξ, η, θ) dξ dη. (11)

A variation δθ produces a variation in dS given by

δ(dS) =1

H

∂H

∂θδθ dS. (12)

The corresponding variation in the cost functional as defined by Equation (10) is

δF =

∫BS

(∂f

∂φδφ +

∂f

∂θδθ +

f

H

∂H

∂θδθ

)dS

+

∫FS

(∂g

∂φδφ +

∂g

∂θδθ +

g

H

∂H

∂θδθ

)dS.

(13)

The terms containing δφ in Equation (13) show that the gradient dF/dθ requires solvingthe field equations N times for a variation in each of the control parameters and that canbe very expensive when the number of these parameters is large. We can replace the δφterms with terms that depend only on the solution to the adjoint problem which is definedand solved only once on the unperturbed geometry. The rest of the terms in Equation (13)depend on the flow solution φ that is also calculated on the same geometry.

11


Let’s rewrite the conditions Equations (2–4) as

B(θ, φ) =∂φ

∂n+ U∞ · n = 0 on BS (14a)

M(θ, φ) =∂2φ

∂x2+ κ

∂φ

∂z= 0 on FS (14b)

and

A(θ, φ) = φ = 0 on FFS (14c)

We note that Equation (1) and conditions Equations (14a) play the role of constraints on thevelocity potential φ, and an admissible variation δφ must be consistent with these constraints.The problem of minimizing F with respect to θ while φ satisfies the respective constraintscan be replaced by unconstrained minimization of a Lagrangian L defined by

L =

∫BS

f dS +

∫FS

g dS +

∫Ω

ψ∇2φ dΩ +

∫BS

βB dS +

∫FS

γM dS +

∫FFS

αA dS, (15)

where ψ, β, γ and α are Lagrange multipliers (also called co-states or adjoint variables) thatare defined on Ω, BS, FS and FFS, respectively. The radiation condition Equation (5),which is also a constraint on φ, is not included in the Lagrangian, and therefore an admissiblevariation δφ must satisfy this condition. Observing that the Lagrangian is a sum of integralseach of the general form

∫G(θ, φ), we write the variation in L due to a variation in θ as

δL =∑ ∫

∂G

∂φ

∣∣∣∣θ

δφ +∑ ∫

∂G

∂θ

∣∣∣∣φ

δθ. (16)

We note that the variations in L due to variations in the Lagrange multipliers all vanishbecause of the constraints on φ (Soemarwoto, 1997). The first term in Equation (16) isthe undesirable term because it requires δφ. This term can be eliminated if we define theLagrange multipliers so that∑ ∫

∂G

∂φ

∣∣∣∣θ

δφ = 0, (17)

and obtain

dLdθ

=∑ ∫

∂G

∂θ

∣∣∣∣φ

. (18)

To this end, we use Green’s identity∫Ω

(ψ∇2φ − φ∇2ψ

)dΩ =

∮Σ

(ψ

∂φ

∂n− φ

∂ψ

∂n

)dS, (19)

12


and write the Lagrangian as

L =

∫BS

f dS +

∫FS

g dS +

∫Ω

φ∇2ψ dΩ +

∫BS

(ψ

∂φ

∂n− φ

∂ψ

∂n+ βB

)dS

+

∫FS

(ψ

∂φ

∂n− φ

∂ψ

∂n+ γM

)dS +

∫FFS

(ψ

∂φ

∂n− φ

∂ψ

∂n+ αA

)dS.

(20)

The variation δL due to δφ is

δL|θ =

∫Ω

δφ∇2ψ dΩ +

∫BS

(∂f

∂φδφ + ψ

∂δφ

∂n− δφ

∂ψ

∂n+ β

∂B

∂φδφ

)dS

+

∫FS

(∂g

∂φδφ + ψ

∂δφ

∂n− δφ

∂ψ

∂n+ γ

∂M

∂φδφ

)dS

+

∫FFS

(ψ

∂δφ

∂n− δφ

∂ψ

∂n+ α

∂A

∂φδφ

)dS.

(21)

The idea is to define ψ, β, γ and α such that each of the integrals in Equation (21) vanishes.It is easy to get rid of the volume integral by defining ψ as a solution to Laplace’s equation

∇2ψ = 0 in Ω. (22)

Taking the variation of Equation (14a), we get

∂B

∂φ

∣∣∣∣θ

δφ =∂δφ

∂non BS, (23)

and we can make the integral on BS vanish by choosing

∂ψ

∂n=

∂f

∂φon BS (24)

and

β = −ψ on BS. (25)

Similarly we note from Equation (14c) that

∂A

∂φ

∣∣∣∣θ

= 1 on FFS, (26)

and the corresponding integral vanishes by setting

ψ = 0 on FFS (27)

13


and

α =∂ψ

∂non FFS. (28)

Equations (24) and (27) are boundary conditions to be satisfied by the solution ψ to Laplace’sequation.

The only integral remaining is that over the free surface. Again we want to choose the freesurface boundary condition on ψ and define γ such that this integral vanishes for arbitraryδφ. Let

IFS =

∫FS

(∂g

∂φδφ + ψ

∂δφ

∂n− δφ

∂ψ

∂n+ γ

∂M

∂φδφ

)dS. (29)

In the linearization used here, the free surface conditions are transferred to z = 0, and hence∂/∂n = ∂/∂z. Recalling the definition of M from Equation (14b), we have

∂M

∂φ

∣∣∣∣θ

δφ =∂2δφ

∂x2+ κ

∂δφ

∂zon FS (z = 0). (30)

Substituting Equation (30) into Equation (29), we get

IFS =

∫FS

[(ψ + κγ)

∂δφ

∂z+

(∂g

∂φδφ − δφ

∂ψ

∂z+ γ

∂2δφ

∂x2

)]dS. (31)

We choose

γ = −ψ

κon FS (z = 0). (32)

This condition defines the Lagrange multiplier γ in terms of ψ on the free surface. The freesurface integral now becomes

IFS =

∫FS

(∂g

∂φδφ − δφ

∂ψ

∂z− ψ

κ

∂2δφ

∂x2

)dS. (33)

The case of a fully submerged body is easier to analyze than the case of a body thatpierces the free surface. In the former case, the body may be shallowly submerged. Weconsider this case first and write the free surface integral as

IFS =

∫ y0

−y0

dy

∫ xu

xd

dx

(∂g

∂φδφ − δφ

∂ψ

∂z− ψ

κ

∂2δφ

∂x2

), (34)

where xd and xu are x-coordinates of the downstream and upstream boundaries of a rectangu-lar domain that covers the free surface, and −y0 and y0 are its boundaries in the y-direction.

14


We integrate the last term by parts;

I1 =

∫ y0

−y0

dy

∫ xu

xd

dxψ∂2δφ

∂x2

=

∫ y0

−y0

dy

[ψ

∂δφ

∂x

∣∣∣∣xu

xd

−∫ xu

xd

dx∂ψ

∂x

∂δφ

∂x

]

=

∫ y0

−y0

dy

(ψ

∂δφ

∂x− δφ

∂ψ

∂x

) ∣∣∣∣xu

xd

+

∫ y0

−y0

dy

∫ xu

xd

dx∂2ψ

∂x2δφ. (35)

Substituting Equation (35) into Equation (34), we obtain

IFS =

∫FS

δφ

(∂g

∂φ− ∂ψ

∂z− 1

κ

∂2ψ

∂x2

)dS −

∫ y0

−y0

1

κ

(ψ

∂δφ

∂x− δφ

∂ψ

∂x

) ∣∣∣∣xu

xd

dy. (36)

We get rid of the first integral by specifying the free surface boundary condition on ψ to be

∂2ψ

∂x2+ κ

∂ψ

∂z= κ

∂g

∂φon FS (z = 0). (37)

In the second integral we invoke the radiation condition Equation (5) which requires thatδφ and ∂δφ/∂x both to be zero as xu → +∞. Hence the integrand of the second integralin Equation (36) vanishes at xu without imposing any conditions on ψ; that is to say ψand ∂ψ/∂x are unconstrained on the upstream boundary. At the downstream boundary φand∂φ/∂x are not constrained because waves may propagate out of the domain across thatboundary. Therefore, we impose two conditions on ψ at the downstream boundary,

ψ and∂ψ

∂x→ 0 as xd → −∞. (38)

This is a radiation condition to be satisfied by the adjoint function ψ. We can imaginethat the adjoint problem is a fictitious flow field in which waves are not allowed downstream,but they may exist upstream. This is opposite to the physical picture of the real flow fieldgiven by the φ problem. Nonetheless, this result is consistent with the use of Rayleigh’smethod of artificial viscosity for enforcing the radiation condition (Kostyukov 1968, page20). If the term −µ∂φ/∂x, with µ > 0, is added to the left side of the free-surface conditionEquation (3), we obtain the term µ∂ψ/∂x in the left side of the adjoint free-surface conditionEquation (37). The change of sign of the artificial viscosity term has implications regardingthe way contours are deformed in the complex wavenumber plane when Fourier integralsare inverted (Kostyukov 1968, page 58). Instead of contours being deformed around thepoles below (or above) the real axis when solving for φ, they are deformed in the oppositedirections when solving for ψ. This results in waves to be present on the upstream boundarybut not downstream for the adjoint.

15


In summary, the adjoint problem for the objective functional Equation (10) is definedby:

∇2ψ = 0 in Ω, (39a)

∂ψ

∂n=

∂f

∂φon BS, (39b)

∂2ψ

∂x2+ κ

∂ψ

∂z= κ

∂g

∂φon FS, (39c)

ψ = 0 on FFS, (39d)

and the radiation condition

ψ and∂ψ

∂x→ 0 as x → −∞. (39e)

The remaining Lagrange multipliers are given in terms of ψ by

β = −ψ on BS, (40a)

γ = −ψ

κon FS (z = 0), (40b)

and

α =∂ψ

∂non FFS. (40c)

With these definitions of the Lagrange multipliers, the gradient dL/dθ is given by Equa-tion (18) which does not require δφ. Of course the gradient depends on the Lagrangemultipliers which have to be determined first by solving the adjoint problem.

Next we consider the case of a surface ship. The task is to specify conditions on ψ on thefree surface so that the integral IFS given by Equation (29) vanishes for δφ that is consistentwith the radiation condition but otherwise arbitrary. The only difference between this caseand the case of a fully submerged body is that integration by parts of the second derivativeterm in Equation (34) produces line integrals on the waterline. We show in Figure 4 arectangular domain that covers the free surface and depict the waterline where the bodypierces the free surface. A line parallel to the x-axis may intersect the waterline at twopoints xB(y) and xS(y) (B and S signify bow and stern, respectively). If the line doesnot intersect the waterline, we take xB = xS. Now we reconsider Equation (35). Afterintegration by parts the integral I1 becomes

I1 =

∫FS

δφ∂2ψ

∂x2dS +

∫ y0

−y0

dy

(ψ

∂δφ

∂x− δφ

∂ψ

∂x

) ∣∣∣∣xu

xd

+

∫ y0

−y0

dy

(ψ

∂δφ

∂x− δφ

∂ψ

∂x

) ∣∣∣∣xS

xB

. (41)

We see that the first two integrals are identical to the integrals of the fully submerged body,and thus we combine them with other terms in IFS to obtain the same free-surface boundary

16


conditions on ψ as those for the submerged case; specifically, Equations (37–38). However,the free surface integral is now reduced to

IFS = −∫ b

−b

1

κ

(ψ

∂δφ

∂x− δφ

∂ψ

∂x

) ∣∣∣∣xS

xB

dy, (42)

where we reduced the limits of integration to ±b, where b is half the maximum beam. Thesubstitution dy = t dx, where t is the slope of the waterline, enables us to integrate the firstterm by parts. The result is

IFS = −1

κ[(tψδφ)u − (tψδφ)l]

∣∣∣∣x2

x1

+

∫ x2

x1

1

κδφ

(ψ

dt

dx+ 2t

∂ψ

∂x

)u

dx

−∫ x2

x1

1

κδφ

(ψ

dt

dx+ 2t

∂ψ

∂x

)l

dx,

(43)

where the subscripts u and l denote the waterline branches (y ≥ 0) and (y ≤ 0), respectively,and x1 and x2 are the trailing and leading edges of the waterline.

The velocity potential φ is unconstrained on the waterline, hence the integral in Equa-tion (42) vanishes if we set ψ = ∂ψ/∂x = 0 on the waterline. We note that if the slope t andcurvature dt/dx both vanish on a portion of the waterline (for example on a long uniform midsection) then the contribution from that portion to the integrals in Equation (43) vanisheswithout putting any constraints on ψ on that portion. However, if the slope of the waterlinedoes not vanish, the only way to make the integrals vanish is to set ψdt/dx + 2t∂ψ/∂x = 0.In this report, we use ψ = 0 if |t| > ε (a small number) otherwise a natural spline conditionin the direction transverse to the waterline is used. The natural spline condition is used inthe SWAN-v2.2 code for the φ problem.

Alternative Forms of the Cost Functional

Boundary condition Equation (24) implies that the integrand of the cost functional f isan explicit function of φ. As an example

f(θ, φ) =1

2(φ − φd)

2, (44)

where φd is a prescribed velocity potential distribution on the hull surface. In this case thehull boundary condition is

∂ψ

∂n= φ − φd on BS. (45)

A more practical situation is a case where f is a function of pressure P [= (p − p∞)/ρ]which is determined from Bernoulli’s equation as

P =1

2U2∞ − 1

2V 2 − g0z. (46)

17


xBxS

x

y

yo

-yo

x1x2

xd xu

Water line

Figure 4. A sketch of free surface boundaries and waterline.

18


We recall that the origin of boundary condition Equation (24) is the requirement that∫BS

δφ∂ψ

∂ndS =

∫BS

δφ∂f

∂φdS. (47)

When f is a function of P , we first write

∂f

∂φδφ =

∂f

∂PδP. (48)

The next step is to write δP in terms of δφ. To this end we introduce a three-dimensionalcurvilinear coordinate transformation

x = x(ξ, η, ζ) (49a)

y = y(ξ, η, ζ) (49b)

z = z(ξ, η, ζ). (49c)

In this system the body surface is given by ζ = 0. The Cartesian velocity componentsare

u =∂φ

∂x=

∂φ

∂ξξx +

∂φ

∂ηηx +

∂φ

∂ζζx (50a)

v =∂φ

∂y=

∂φ

∂ξξy +

∂φ

∂ηηy +

∂φ

∂ζζy (50b)

w =∂φ

∂z=

∂φ

∂ξξz +

∂φ

∂ηηz +

∂φ

∂ζζz. (50c)

The variation in the velocity components due to a variation δφ are

δu =∂δφ

∂ξξx +

∂δφ

∂ηηx +

∂δφ

∂ζζx (51a)

δv =∂δφ

∂ξξy +

∂δφ

∂ηηy +

∂δφ

∂ζζy (51b)

δw =∂δφ

∂ξξz +

∂δφ

∂ηηz +

∂δφ

∂ζζz. (51c)

To determine δP , we use

V 2 =(U∞ + u

)2+

(V ∞ + v

)2+

(W∞ + w

)2(52)

V 2∞ = U

2

∞ + V2

∞ + W2

∞, (53)

where U∞, V ∞ and W∞ are the Cartesian velocity components in the far field. In thepresent work, we have: U∞ = −U∞ and V ∞ = W∞ = 0. It follows from Equation (46) that

P = −(U∞u + V ∞v + W∞w

)− 1

2

(u2 + v2 + w2

)− g0z. (54)

19


Taking the variation of this equation, we obtain

δP = −[(

U∞ + u)δu +

(V ∞ + v

)δv +

(W∞ + w

)δw

]. (55)

Substituting for the variations in the velocity components from Equations (51a–51c), we get

δP = −(

U∂δφ

∂ξ+ V

∂δφ

∂η+ W

∂δφ

∂ζ

), (56)

where U , V and W are the contravariant velocity components:

U =(U∞ + u

)ξx +

(V ∞ + v

)ξy +

(W∞ + w

)ξz (57a)

V =(U∞ + u

)ηx +

(V ∞ + v

)ηy +

(W∞ + w

)ηz (57b)

W =(U∞ + u

)ζx +

(V ∞ + v

)ζy +

(W∞ + w

)ζz. (57c)

The no-penetration condition Equation (2) can be written exactly as W = 0, thus

δP = −(

U∂δφ

∂ξ+ V

∂δφ

∂η

)on BS. (58)

With δP given by Equation (58), substitution of Equation (48) into Equation (47) gives∫BS

δφ∂ψ

∂ndS = −

∫BS

∂f

∂P

(U

∂δφ

∂ξ+ V

∂δφ

∂η

)dS. (59)

Recalling the expression of the surface element Equation (11) and assuming that the bodysurface is mapped into a rectangular domain in the plane ζ = 0, we write Equation (59) as∫

BS

δφ∂ψ

∂ndS = −

∫ η2

η1

∫ ξ2

ξ1

∂f

∂PHU

∂δφ

∂ξdξ dη −

∫ η2

η1

∫ ξ2

ξ1

∂f

∂PHV

∂δφ

∂ηdξ dη, (60)

where ξ1, ξ2, η1, and η2 are the boundaries of the rectangular domain. Integration by partsyields ∫

BS

δφ∂ψ

∂ndS =

∫BS

δφ1

H

[∂

∂ξ

(HU

∂f

∂P

)+

∂

∂η

(HV

∂f

∂P

)]dS

−∫ η2

η1

δφHU∂f

∂P

∣∣∣∣ξ2ξ1

dη −∫ ξ2

ξ1

δφ HV∂f

∂P

∣∣∣∣η2

η1

dξ.

(61)

For a fully submerged body, the η-line integral vanishes because ξ1 and ξ2 correspondto singular lines at the nose and tail points. Also, the ξ-line integral vanishes because ηwould be a periodic direction (or η1 and η2 are on a plane of symmetry), hence the integrand

20


has equal values on η1 and η2 (or zero in the case of plane of symmetry). It follows fromEquation (61) that the boundary condition on ψ is

∂ψ

∂n=

1

H

[∂

∂ξ

(HU

∂f

∂P

)+

∂

∂η

(HV

∂f

∂P

)]on BS. (62)

For the objective functionals Equations (6) and (9), we find ∂f/∂P to be −ρnx and ρ(P−Pd),respectively. The first case is for minimizing wave resistance and the second is for matchinga target pressure distribution on the hull.

Next we consider the case where the function g in the cost functional Equation (10)depends on the free-surface elevation η0 above the undisturbed water surface. We use thelinearized equation

η0 =U∞g0

u =U∞g0

∂φ

∂x. (63)

Following similar procedure as that for the hull boundary condition, we replace the freesurface condition Equation (39c) by

∂2ψ

∂x2+ κ

∂ψ

∂z= − 1

U∞

∂

∂x

(∂g

∂η

)on FS (z = 0). (64)

The cost functional Equation (7) which gives the wave resistance as an integral on a yz-planein the ship’s wake is more reliable than integrating the pressure on the hull surface. We haveformulated the adjoint problem for this cost functional. It is given by Equations (39a–39d)with homogeneous conditions on BS and FS (f = 0 and g = 0), but with non-homogeneousconditions on the downstream plane WS: ψ = −ρ∂φ/∂x and ∂ψ/∂x = −ρ∂2φ/∂x2.

The analysis leading to Equation (62) may imply that a three-dimensional curvilin-ear grid is needed. Such a grid would be available if a field solver such as finite vol-ume/difference/element is used for solving the flow equations. However, panel methodsare commonly used for potential flows. Equation (62) can still be implemented in a panelcode without generating a three-dimensional grid. In a panel code, a surface grid is natu-rally available which gives the coordinates of a surface point by a parameterization [xB(ξ, η),yB(ξ, η), zB(ξ, η)], where ξ and η are two surface parameters. A volume parameterization inthe neighborhood of the surface BS can be constructed as follows

x = xB(ξ, η) − ζnx (65a)

y = yB(ξ, η) − ζny (65b)

z = zB(ξ, η) − ζnz, (65c)

where (nx, ny, nz) is the outward unit normal and ζ is the distance along the normal. Thebody surface is given by ζ = 0. The derivatives needed in Equation (62) can be found from

∂(x, y, z)

∂(ξ, η)

∣∣∣∣∣ζ=0

=∂(xB, yB, zB)

∂(ξ, η)(66)

21


and

∂(x, y, z)

∂ζ

∣∣∣∣∣ζ=0

= −(nx, ny, nz). (67)

The Gradient dL/dθ

We can evaluate the gradient dL/dθ after solving the adjoint problem and determiningthe Lagrange multipliers. To this end we develop two auxiliary relations (Soemarwoto 1997).First, consider a volume integral

F1 =

∫Ω

f1(θ, φ) dΩ. (68)

The gradient of F1 for fixed φ is

∂F1

∂θ=

∫Ω

∂f1

∂θdΩ +

∫Ω

f1∂

∂θ(dΩ). (69)

The derivative of the volume element with respect to θ is

∂

∂θ(dΩ) =

∂

∂θ(dx) dy dz +

∂

∂θ(dy) dx dz +

∂

∂θ(dz) dx dy. (70)

Introducing the grid velocity

ω = (ωx, ωy, ωz) =∂x

∂θı +

∂y

∂θ +

∂z

∂θk, (71)

we write Equation (70) as

∂

∂θ(dΩ) = d(ωx) dy dz + d(ωy) dx dz + d(ωz) dx dy

=

(∂ωx

∂x+

∂ωy

∂y+

∂ωz

∂z

)dx dy dz

or

∂

∂θ(dΩ) = ∇ · ω dΩ, (72)

which reminds us of the fact that the rate of change of volume per unit of volume is thedivergence of the velocity field.

With the help of ω we can write

∂f1

∂θ= ω · ∇f1. (73)

22


Substituting Equations (72) and (73) into Equation (69), we obtain

∂F1

∂θ=

∫Ω

(ω · ∇f1 + f1∇ · ω)dΩ

=

∫Ω

∇ · (f1ω) dΩ (74)

Finally, using Green’s theorem, we have

∂F1

∂θ=

∫Σ

f1ωn dS, (75)

where ωn is the projection of ω onto the normal to the boundary of Ω.

The second auxiliary relation is the gradient of a surface integral

F2 =

∫Σ

f2(θ, φ) dS. (76)

Recalling Equations (11) and (12), we write the gradient of F2 at fixed φ as

∂F2

∂θ=

∫Σ

1

H

∂f2H

∂θdS. (77)

Applying Equations (75) or (77) appropriately to terms in Equation (15), we get

dLdθ

=

∫BS

1

H

∂fH

∂θdS +

∫Σ

ωnψ∇2φ dS +

∫BS

1

H

∂

∂θ(βHB) dS

+

∫FS

1

H

∂

∂θ(gH) dS +

∫FS

1

H

∂

∂θ(γHM) dS +

∫FFS

1

H

∂

∂θ(αHA) dS.

(78)

Recalling that ∇2φ = 0 in Ω including points on its boundary and imposing the constraintsEquations (14a–14c), we obtain

dLdθ

=

∫BS

(1

H

∂fH

∂θ+ β

∂B

∂θ

)dS +

∫FS

(1

H

∂gH

∂θ+ γ

∂M

∂θ

)dS +

∫FFS

α∂A

∂θdS. (79)

On FFS we have A = φ = 0, hence ∂A/∂θ = 0. Using Equations (25) and (32) for β andγ, we write Equation (79) as

dLdθ

=

∫BS

(1

H

∂fH

∂θ− ψ

∂B

∂θ

)dS +

∫FS

(1

H

∂gH

∂θ− ψ

κ

∂M

∂θ

)dS. (80)

This is the desired result. It can be shown that ∂M/∂θ = 0 on the free surface. The gradientis composed of two kinds of terms, the first gives the change in the cost functional due togeometry variations while the flow field is fixed. The second, which is now given in terms ofthe solution to the adjoint problem ψ, accounts for the change in flow field δφ. For a deeply

23


submerged sphere, analytical solutions to the two problems for φ and ψ can be determined.This model problem can be used to show that the two kinds of terms are equally important.

For a submerged body, g and H being defined on the free surface cannot depend explicitlyon θ. Hence ∂gH/∂θ = 0 on FS and the free-surface integral in Equation (80) vanishes. Itis interesting to note that if f = 0 but g = 0, the gradient depends only on the term ψ∂B/∂θwhich has no explicit reference to the cost functional. The function g affects the gradientimplicitly through the free-surface condition on ψ as shown by Equation (39c).

Dawson’s Linearized Free-Surface Condition

In the Dawson’s formulation of the ship problem, the basis flow, around which the freesurface conditions are linearized, is the double-body flow. Nakos and Sclavounos (1990)provide a rigorous treatment of the double-body linearization and their formulation for steadywaves is used in this section. The velocity potential is decomposed into three components:the incoming uniform stream −U∞x, the double-body disturbance potential φ, and thesteady wave potential φw. The sum of the first two potentials gives the double-body flow.Two problems are solved in succession. The first is for φ where the no-penetration conditionis enforced on the hull surface BS and the undisturbed free surface FS as if it were rigid.Next the φw problem is solved with a linearized free-surface condition whose coefficientsnow depend on the double-body solution φ. The double-body solution also appears as anon-homogeneous term in the hull surface boundary conditions.

It appears that two adjoint problems are needed: one for φ and another for φw. Here,the linear problem for φw is rewritten in terms of the total disturbance potential φ = φw +φ.The advantage of working with φ is that φ appears only in the coefficients of the free-surfacecondition, which is still linear in φ, but it does not appear in the hull surface condition.In fact the governing equations for φ are Equations (1, 2, 4, and 5) and the free-surfacecondition:

A∂2φ

∂x2+ B

∂2φ

∂y2+ C

∂2φ

∂x∂y+ D

∂φ

∂x+ E

∂φ

∂y+ g0

∂φ

∂z− F = 0 on FS (z = 0), (81)

24


where

A =

(−U∞ +

∂φ

∂x

)2

(82a)

B =

(∂φ

∂y

)2

(82b)

C = 2

(−U∞ +

∂φ

∂x

)∂φ

∂y(82c)

D = 3

(−U∞ +

∂φ

∂x

)∂2φ

∂x2+ 2

∂φ

∂y

∂2φ

∂x∂y+

(−U∞ +

∂φ

∂x

)∂2φ

∂y2(82d)

E = 3∂φ

∂y

∂2φ

∂y2+ 2

(−U∞ +

∂φ

∂x

)∂2φ

∂x∂y+

∂φ

∂y

∂2φ

∂x2(82e)

F =1

2

[5

(−U∞ +

∂φ

∂x

)2

+

(∂φ

∂y

)2

+ U2∞

]∂2φ

∂x2

+1

2

[(−U∞ +

∂φ

∂x

)2

+ 5

(∂φ

∂y

)2

+ U2∞

]∂2φ

∂y2

+ 4

(−U∞ +

∂φ

∂x

)∂φ

∂y

∂2φ

∂x∂y.

(82f)

In deriving the expression of F , the boundary condition ∂φ/∂z = 0 on the free surface isused.

The double-body disturbance potential φ does not appear explicitly anywhere in thegoverning equations of the total disturbance potential φ except in the coefficients of the free-surface condition Equation (81). Therefore, a perturbation in geometry produces variationsδφ and δφ, the second of them affects only the mentioned coefficients. Solving an adjointequation for the double-body potential is avoided by neglecting variations in the coefficientsof the free-surface condition. With this approximation in mind, the derivation presentedpreviously starting on page 11 for the Kelvin’s free surface condition is repeated for thedouble-body linearization. The adjoint problem is still defined by Equations (39a) and (39e)except that the free surface condition Equation (39c) is now replaced by:

∂2Aψ

∂x2+

∂2Bψ

∂y2+

∂2Cψ

∂x∂y− ∂Dψ

∂x− ∂Eψ

∂y+ g0

∂ψ

∂z= g0

∂g

∂φon FS (z = 0). (83)

This condition can be adapted to the SWAN-v2.2 code.

25


Submarines Operating Near a Free Surface

In this section, we demonstrate the use of the adjoint formulation for shape optimizationof submarine configurations operating near a free surface. We seek hull forms of minimumwave resistance subject to certain geometric constraints that will be given later. We alsosolve the inverse problem in which a target pressure distribution on hull is specified and ahull form is sought. We use double-body linearization of the free-surface conditions in allresults in this section unless otherwise noted.

Geometry Parameterization by B-Spline Surfaces

A method of geometry parameterization that allows systematic variations in shape isessential for shape optimization. Such a method is B-spline surfaces (Mortenson 1997). Thetensor product equation is

r(ξ, η) =m∑

i=0

n∑j=0

rijNi,K(ξ)Nj,L(η), (84)

where rij are position vectors of the control points. The Ni,K(ξ) and Nj,L(η) are the basisfunctions. We use a uniform bi-cubic B-spline surface with C2 continuity. The ξ-parameterruns along a meridional line from bow to stern and the η-parameter runs on a station line(x = constant) from keel to crown. We assume that the submerged body has a plane ofsymmetry. There are (m + 1)(n + 1) control points on half the body and each control pointis identified by two indices (i, j) where i = 0, 1, . . . , m and j = 0, 1, . . . , n. For later use, wecombine the two indices into a one dimensional index k = (m + 1)j + i + 1. These indicesare depicted in Figure 5 for the case m = 8 and n = 14. The streamwise positions xij of thecontrol points vary only with index j, however, their offsets yij and depths zij vary with bothi and j. In the rest of this section, we identify a control point by the index k. The streamwisecoordinates xk of the control points are held fixed while the offsets yk and depths zk can beused as design variables. Because of symmetry and other constraints on the depth of thenose and tail points, the number of control parameters is reduced to (m − 3)(n − 3) offsetsand (m − 3)(n − 1) depths. We note that the number of panels used to represent the bodyin the flow solver SWAN-v2.2 is much larger than the number of B-spline surface patches.There are (m− 2)(n− 2) surface patches on half the body and each patch is subdivided into(mp×np) panels. The total number of panels is mpnp(m−2)(n−2). Grid sensitivity studiescan be easily achieved by varying one or more of the four parameters (m, n, mp, np).

Base-Line Submarine G0

A base-line submarine, which is denoted by G0, is given by a composite B-spline surfacepatches. The control points are placed on a mathematically defined surface, however, thissurface is used only for the definition of G0 and is not needed for the optimization procedure.The bow region is a semi-ellipsoid with semi-axes a = 0.15 and b = 0.06 and the stern is

26


also a semi-ellipsoid with a = 0.30 and b = 0.06. These semi-ellipsoids are connected bya cylindrical section of length of 0.55 and radius of 0.06. The base-line hull G0, which iscontained within this mathematically defined surface, is nearly a body of revolution. Thelength of G0 is re-scaled to be one and its nominal diameter is 0.1146, hence L/D = 8.73.

For m = 8 and n = 14 the B-spline control points indices are depicted in Figure 5. Thenose point corresponds to the line j = 1, i = 1, . . . , 7, and the tail point to the line j = 13,i = 1, . . . , 7. The keel corresponds to the line i = 1, j = 1, . . . , 13, and the crown line tothe line i = 7, j = 1, . . . , 13. Each patch is subdivided into (mp × np) = (5 × 6) panelswhich gives 73 stations and 31 waterlines on half the hull. A projection of G0 on the planeof symmetry y = 0 is depicted in Figure 6. The axis depth is z = −0.1 and the nose andtail points are placed at x = 0.5 and x = −0.5, respectively. The displacement of G0 isV0 = 0.00866 and its wetted surface area is S0 = 0.327.

The flow around G0 is solved using SWAN-v2.2 code. The basis flow around which thefree-surface conditions are linearized is the double-body flow (similar to Dawson’s lineariza-tion, see Nakos and Sclavounos (1990) for more details). The wave resistance coefficient asa function of Froude number (based on ship length) is shown in Figure 7. There are twocurves corresponding to two different methods of calculating wave resistance in SWAN-v2.2code. The curve denoted by Rw is based on transverse wave cut method while Rp is obtainedby integrating hull surface pressure.

Wave Resistance Minimization at Froude Number Fn = 0.230

The wave resistance of G0 has local maxima near Froude numbers Fn = 0.230 and 0.300.We choose to optimize the hull at these Froude numbers subject to geometric constraints.Here we consider the lower Froude number. The objective functional is the wave resistanceas defined by Equation (6). The design variables are the offsets of the B-spline control pointswhile their depths and streamwise locations are held fixed. However, not all of the controlpoints are available as design variables. This is because the first station (nose point), the laststation (tail point), the keel and crown must all have zero offsets. The control points whichare free to change correspond to indices i = 2, . . . , m − 2 and j = 2, . . . , (n − 2). Therefore,the number of design variables is ndv = (m − 3)(n − 3) which is 55 in the present example.

Since the depths and streamwise positions of the control points are fixed, the optimizedhull, which is denoted by Gd sub230, will have a profile in the plane of symmetry as that ofthe original hull G0. Two constraints are imposed on Gd: the displacement of Gd satisfies anequality constraint Vd = V0, whereas its wetted surface area satisfies an inequality constraintSd ≤ 1.3 S0. This means that the displacement of the optimized hull must be equal to thatof the original hull but its surface area may be greater than the original surface area.

We must also specify side constraints. To this end, we let y0k denote the initial values of the

design variables that give the original hull G0, and let d1 denote the minimum of these values.We specify side constraints by: d1 ≤ yk ≤ 0.1. This constrained optimization problem issolved using a generalized reduced gradient method (GRG) as detailed by Belegundu and

27


Chandrupatla (1999, page 176). In this method, inequality constraints are replaced withequality constraints by introducing slack variables.

It is important to establish credibility of the adjoint method for computing the gradient ofthe cost functional with respect to the design variables yk. For this purpose, we compare thegradient found by the adjoint with that by a finite-difference method. In the latter method,we perturb the control variables yk one at a time and recalculate the cost functional oneach perturbed hull. Knowing the cost functional on the unperturbed hull, we compute thegradient by a first-order forward finite-difference method. There are 55 control variables, andthus the flow field has to be recomputed on 55 different hulls in addition to the unperturbedhull. This should be compared with only two solutions for the adjoint approach; one for thevelocity potential and the other for the adjoint function. The 55 components of the gradientcomputed by the two methods are depicted in Figure 8 against the index k which is definedby Figure 5. The overall agreement between the two methods is excellent. However, thereare discrepancies in the nose and tail regions which we attribute to grid stretching and largepressure gradient in these regions. In Figure 9 we depict the difference between the twomethods after normalization by the magnitude of the gradient vector. The error is less than3 percent.

In the first two design cycles, the cost functional is reduced to 0.100 and 0.008 of itsinitial value, respectively. Therefore, the design iteration was terminated. The designedhull, which is denoted by Gd2 sub230, is shown in Figures 10–13. Its wetted surface area is0.332, which is 1.5 percent larger than the original surface area. The major changes in shapehappen in the mid section which is pinched as evident in Figure 13, followed by a smallincrease in beam fore and aft of that section as shown in Figure 12. This redistribution ofsectional area along the body axis results in destructive wave interference which is the onlymechanism available for wave resistance minimization in the present flow model. The wavedrag coefficient (based on wetted surface area) of the base-line hull G0 is Rw = 1.22 × 10−3

and of the designed hull it is 0.0197× 10−3. The wave drag of Gd2 sub230 is about 2 percentof that of G0. The optimized hull is practically a waveless hull at Froude number 0.230. Itis important to recall that this result is based on potential flow theory.

The lower wave resistance of Gd2 sub230 manifests itself in the wave pattern on the freesurface and in the plane of symmetry wave elevation shown in Figures 14 and 15, respectively.From Figure 15 we see that the wave length of transverse waves on the original hull G0 isapproximately 0.333. That is to say, the body length is about three times the wave length.Hence, the local maximum in the wave resistance at Fn = 0.230 is a direct consequenceof constructive wave interference between the bow and stern waves. Pinching the originalhull at its mid section divides the hull into two bodies each with an effective length that is1.5 times transverse wave length resulting in destructive wave interference and hence dragreduction.

It is interesting to investigate the pressure distribution on the hull surface. Top viewsof pressure contours on the two hulls G0 and Gd2 sub230 are shown in Figures 16 and 17,

28


respectively. Side views of the same contours are shown in Figures 18 and 19. The footprintsof surface waves are clearly visible on the top of G0 sub230, with a suction region on thestern induced by a trough above it. On the designed hull Gd2, pressure contours are similarto those on G0 only on bow, but significantly different on stern where the suction regionis replaced by higher pressure induced by a crest above the stern. The removal of the lowpressure region on the stern is the source of drag reduction on the designed hull.


We noted earlier that the wave resistance of the base-line submarine G0 shown in Figure 7has strong local maximum at Froude number Fn = 0.300. In this section, we minimize waveresistance at this Froude number. The constrained optimization problem is exactly the sameas described in the previous section except for the Froude number.

At Froude number Fn = 0.300, we first establish credibility of the adjoint method forcomputing the gradient of the cost functional with respect to the design variables yk. Againwe compare with a first-order finite-difference method. The 55 components of the gradientvector computed by the two methods are depicted in Figure 20 against the index k whichis defined by Figure 5. The overall agreement between the two methods is excellent. InFigure 21 we depict the difference between the two methods after normalization by themagnitude of the gradient vector. Here, the error is also less than 3 percent.

The reduction of the cost functional Equation (6) obtained by GRG method is shown inFigure 22. The iteration process is terminated after 10 design cycles since the reduction isdeemed sufficient while further reduction becomes very slow in the current GRG method. Infact, the cost functional is reduced to 0.12 of its initial value in the first five cycles.

The designed hull, which is denoted by Gd10 sub300, is shown in Figures 23–27. Thisshape is characteristically different from the one obtained at the lower Froude number ofFn = 0.230. Its bow and stern are thinner than those of G0 whereas its mid section isfattened and flattened in the horizontal direction thereby keeping equal displacement asG0. This is an attempt by the hull to shorten its effective length so that destructive waveinterference can be realized. The side constraints on the design variables play an importantrole in this case. They limit the extent by which the body expands or shrinks in the horizontaldirection. The wetted surface area of Gd10 sub300 is 0.333, which is 1.8 percent larger thanthat of G0.

It may be anticipated that a minimum wave resistance body should be symmetric foreand aft of the mid section. Such symmetry is lacking in the current design because the planeof symmetry profile, which is held fixed, is not symmetric fore and aft of mid section. Also,double-body linearization breaks the fore-aft symmetry.

The wave resistance coefficient of Gd10 sub300 over a range of Froude number is comparedwith that of G0 in Figure 28. At the optimization Froude number Fn = 0.300, the waveresistance coefficient of G0 is Rw = 5.37 × 10−3 and for the optimized hull Gd10 sub300 itis Rw = 0.0878 × 10−3. It is important to recall that this result is based on potential flow

29


theory. We also note that the wave resistance of the designed hull is lower than that of G0

over the Froude number range 0.26 to 0.34; for higher Froude numbers this trend is reversed.

The minimum wave resistance of Gd10 sub300 manifests itself in the free surface wavepattern and in the plane of symmetry wave elevation shown in Figures 29 and 30, respectively.From Figure 30 we see that the wave length of transverse waves on the original hull G0

is approximately 0.54. That is to say the original body length is about twice transversewave length. Hence, the local maximum in the wave resistance at Fn = 0.300 is a directconsequence of constructive wave interference between the bow and stern waves. The trickof pinching the original hull at its mid section does not work at this higher Froude numberbecause the resulting body would have an effective length equal to one transverse wavelength. The fact that the shape of the optimized hull Gd10 sub300 is thinner at bow andstern but fatter in the middle suggests that its characteristic length of relevance to waveinterference is actually less than 1; but it must be greater than 0.5 to cause destructiveinterference. The basic features of the wave pattern of Gd10 sub300 shown in Figures 29and 30 are two crests above bow and stern separated by a trough above mid section. Thesefeatures are similar to those of the pressure coefficient distribution that would exist on thefree surface as if it were a rigid lid (a body in ground effect).

It is interesting to investigate the pressure distribution on the hull surface. The pressuredistribution on bow is shown in Figure 31 and on stern in Figure 32. The left side of eachfigure is for G0 and the right side for Gd10 sub300. A thinner nose results in a smaller regionof positive Cp and a larger suction region on the forward half of hull. The major changesin the pressure distribution on stern is the removal of the strong suction that used to existon top of G0 and its replacement with a slightly positive pressure coefficient. The removalof low pressure region on the stern is the source of drag reduction on the designed hull.This conclusion is also supported by Figures 33 and 34. The contributions of small stripsbetween stations x and x + dx to the cost functional Equation (6) is shown as a function ofx in Figure 33. The cumulative effect of all stations between bow and station x is shown inFigure 34. It is interesting to note that the forward half of the optimized hull experienceszero wave drag and only the 3rd quarter of body contributes all the drag. Almost all of thedrag reduction on the designed hull Gd10 sub300 is due to the improved pressure distributionon the last 25 percent of the hull. This is a bit unfortunate because viscous effects andinteractions with propellers dominate the flow field in this region. Considerations of theseeffects and interactions are necessary for the design of the stern region.

Using the Depths of Control Points as Design Variables

In the previous two sections, the offsets of B-spline control points are used as designvariables while their depths are held fixed. In this section, both the offsets and depths ofsubmergence are used as design variables. This introduces more degrees of freedom andflexibility in shape optimization. Only in this section of the report, Kelvin’s rather thandouble-body linearization of the free-surface conditions is used. The only reason for using

30


Kelvin’s linearization is that the results in this section have been obtained in the early stagesof code development.

The accuracy of Equation (80) for calculating the gradient is validated by compari-son with direct calculations using second-order finite differences, dL/dθ ≈ [L(θ + ∆θ) −L(θ − ∆θ)]/2∆θ. To this end, we consider a simple body and use bi-cubic B-spline sur-faces (Mortenson 1997) for parameterization. The control points of the B-spline surfaces areplaced on an ellipsoid (length = L = 1, diameter = 0.2, axis depth = −0.2) and we denotethis body by G0. There are (m+1)(n+1) control points on half the body which is assumedto have a plane of symmetry but the body is not axisymmetric. The streamwise coordinates,xk, of the control points are held fixed while the offsets, yk, and depths, zk, are consideredto be the control parameters θi. The index k takes the values k = 1, 2, . . . , (m + 1)(n + 1).Because of symmetry and other constraints on the depth of the nose and tail points, thenumber of free control parameters is (m − 3)(n − 3) offsets and (m − 3)(n − 1) depths. Inthe results presented here, we use m = 8 and n = 18, which results in a total of 180 controlparameters for half the body. We note that each B-spline patch is subdivided into 4 × 4panels resulting in 1536 panels on half the body.

It is important to note that to determine the gradient by central finite differences, the 180control parameters are perturbed one at a time, and for each parameter the field equationsare solved twice on a perturbed geometry. In contrast, in the adjoint approach the field andadjoint equations are solved only once on unperturbed geometry. Evaluation of the gradientfrom Equation (80) requires geometric properties (area, centroid, unit normal of each panel)of the perturbed hull forms but no field equations are solved. This is a clear advantage ofthe adjoint approach over finite differences.

Wave Resistance Minimization

The first example is for wave resistance minimization whose objective functional is givenby Equation (6). At a Froude number (Fn = U∞/

√g0L) of 0.450, we compare the components

of the gradient by adjoint and finite differences in Tables 1 and 2. The discrepancies are lessthan 3 percent. Only offsets and depths of control points that define the body nearest tothe free surface are shown.

Design for a Target Pressure Distribution

The second example concerns the inverse problem whose objective functional is given byEquation (9). The target pressure pd is the pressure distribution that would result on theB-spline body G0 if we replace the free surface by a rigid lid (also known as the double bodyor zero-Froude-number pressure). This pressure is significantly different from the actualfree-surface flow on the same body at finite Froude number. It is important to note that therigid lid condition is used once at the beginning of calculations only to provide the targetpressure distribution. When solving for φ and ψ, the respective free-surface conditions atfinite Froude number are enforced. At a Froude number of 0.450, Tables 3 and 4 show that

31


the discrepancies between the finite differences and adjoint formulation is less than 5 percent.

In this example, a simple unconstrained steepest descent method is used to determinean optimal geometry for this inverse problem. The offsets, yk, are constrained to be posi-tive. Some of the offsets start to become negative after 108 functional evaluations and thecalculations are terminated. At this stage the cost functional is reduced to 0.02 of its initialvalue. The designed body, Gd sub450 is shown in Figure 35. Significant changes in the crosssections can be observed near the free surface. The cross sections in the forward half of body(0 ≤ x ≤ 0.5L) develop a flat top, whereas sections in the rear half (−0.5L ≤ x ≤ 0) becomethinner near the free surface forming a fin-like structure. However, the cross sections remainnearly circular in the lower half that is farther away from the free surface.

In Figure 36 we depict three pressure coefficient distributions: The target pressure (solidline) which is the pressure distribution on the B-spline body G0 at zero Froude number, thepressure on the optimal solution Gd sub450 (filled circles), and the pressure on G0 at Froudenumber of 0.450 (dashed line). All the distributions are on the top meridional line. Thepressure distribution on Gd sub450 has a lower suction peak and milder adverse pressuregradient than the initial body G0. Because the volume is not constrained, the designed bodyhas a slightly smaller volume. However, it is found that the wave drag per unit volume ofthe designed body is 0.60 that of the initial body G0.

32


Table 1. dF/dyk by adjoint and central finite difference, minimum wave resis-tance, and offsets yk of B-spline control points

k yk Adjoint Finite Diff. |%| discrep

43 +0.27779E−01 −0.34456E−03 −0.35413E−03 2.7052 +0.35355E−01 −0.32731E−03 −0.33133E−03 1.2161 +0.41573E−01 +0.58274E−04 +0.58805E−04 0.9070 +0.46194E−01 +0.80873E−03 +0.81433E−03 0.6979 +0.49039E−01 +0.15918E−02 +0.16021E−02 0.6488 +0.50000E−01 +0.19319E−02 +0.19383E−02 0.3397 +0.49039E−01 +0.15948E−02 +0.15959E−02 0.07

106 +0.46194E−01 +0.79862E−03 +0.79763E−03 0.12

Table 2. dF/dzk by adjoint and central finite difference, minimum wave resis-tance, and depths zk of B-spline control points

k zk Adjoint Finite Diff. |%| discrep

44 −0.14444E+00 −0.42650E−03 −0.43374E−03 1.6753 −0.12929E+00 −0.44952E−03 −0.45543E−03 1.3062 −0.11685E+00 −0.47806E−04 −0.48654E−04 1.7471 −0.10761E+00 +0.82454E−03 +0.82817E−03 0.4480 −0.10192E+00 +0.18179E−02 +0.18245E−02 0.3689 −0.10000E+00 +0.23273E−02 +0.23312E−02 0.1798 −0.10192E+00 +0.19593E−02 +0.19588E−02 0.02

107 −0.10761E+00 +0.93872E−03 +0.93685E−03 0.20

33


Table 3. dF/dyk by adjoint and central finite difference, prescribed surface pres-sure, and offsets yk of B-spline control points

k yk Adjoint Finite Diff. |%| discrep

43 +0.27779E−01 −0.11320E−03 −0.11053E−03 2.4152 +0.35355E−01 −0.14600E−03 −0.14272E−03 2.3061 +0.41573E−01 −0.93342E−04 −0.91620E−04 1.8870 +0.46194E−01 +0.66461E−04 +0.65918E−04 0.8279 +0.49039E−01 +0.26798E−03 +0.26625E−03 0.6588 +0.50000E−01 +0.44116E−03 +0.43780E−03 0.7797 +0.49039E−01 +0.58772E−03 +0.58339E−03 0.74

106 +0.46194E−01 +0.66642E−03 +0.66375E−03 0.40

Table 4. dF/dzk by adjoint and central finite difference, prescribed surface pres-sure, and depths zk of B-spline control points

k zk Adjoint Finite Diff. |%| discrep

44 −0.14444E+00 −0.12224E−03 −0.11911E−03 2.6353 −0.12929E+00 −0.16318E−03 −0.15954E−03 2.2862 −0.11685E+00 −0.11501E−03 −0.11310E−03 1.6971 −0.10761E+00 +0.47383E−04 +0.45716E−04 3.6580 −0.10192E+00 +0.26455E−03 +0.25958E−03 1.9189 −0.10000E+00 +0.49133E−03 +0.48480E−03 1.3598 −0.10192E+00 +0.73054E−03 +0.72554E−03 0.69

107 −0.10761E+00 +0.85242E−03 +0.85207E−03 0.04

34


0 1 2 3 4 5 6 7 8 9 10 11 12 13 140

1

2

3

4

5

6

7

8

i

j

k=

21

22

23

24

25

30 39 48 66 75 84 93 102 11157

115

114

113

112

10643 52 61 70 79 88 9734

Bow Stern

Figure 5. Indices of bi-cubic B-spline control points for a submarine configuration.

35


x

z

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

Figure 6. Base-line submarine G0, z = 0 is the free surface.

Fr

Rw,R

p

0.15 0.2 0.25 0.3 0.35 0.4 0.450

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

Rw

Rp

Figure 7. Wave resistance coefficient for base-line submarine G0 (Rp by pressureintegration and Rw by wave cut method).

36


k20 40 60 80 100 120

-0.0025

-0.0020

-0.0015

-0.0010

-0.0005

0.0000

0.0005

0.0010

0.0015

0.0020

Gradient

AdjointFinite Difference

Figure 8. Gradient computed by adjoint and finite-difference methods, Fn = 0.230.

k

erro

r

20 40 60 80 100 120

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

Figure 9. Errors in gradient components normalized by gradient vector magni-tude, Fn = 0.230.

37


XY

Z

Figure 10. A view of designed hull Gd2 sub230.

38


x

z

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

Figure 11. Projection on the xz-plane of designed hull Gd2 sub230.

x

y

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-0.1

-0.05

0

0.05

0.1

0.15

Figure 12. Projection on the xy-plane of designed hull Gd2 sub230.

39


y

z

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

x = 0

x = 0.25

x = -0.25

Figure 13. Cross sections of designed hull Gd2 sub230.

40


X

Y

-1.5 -1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

0.001

0.001

-0.001

-0.001

Gd2

Go

Figure 14. Wave pattern of base-line hull G0 (lower half) and designed hull Gd2

sub230 (upper half), Fn = 0.230.

41


x

η

-1.5 -1 -0.5 0 0.5 1-0.015

-0.01

-0.005

0

0.005

0.01

0.015Gd2Go

Figure 15. Wave elevation in plane of symmetry of base-line hull G0 and designedhull Gd2 sub230, Fn = 0.230.

42


X

Y

Z

0.0 -0.0090.0-0.008

0.005

Figure 16. Pressure contours (0.5F 2nCp) on top of base-line hull G0, Fn = 0.230.

X

Y

Z

0.0 -0.0080.0

-0.006 0.004

Figure 17. Pressure contours (0.5F 2nCp) on top of designed hull Gd20 sub230, Fn = 0.230.

43


Y X

Z

-0.003

-0.0050.0

-0.006 0.005

Figure 18. Pressure contours (0.5F 2nCp) on side of base-line hull G0 sub230, Fn = 0.230.

Y X

Z

0.0

-0.0050.0

-0.006 0.005

Figure 19. Pressure contours (0.5F 2nCp) on side of designed hull Gd2 sub230, Fn = 0.230.

44


k20 40 60 80 100 120

-0.005

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0.004


Gradient

Figure 20. Gradient computed by adjoint and finite-difference methods, Fn = 0.300.

k

erro

r

20 40 60 80 100 120

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05


45


0 1 2 3 4 5 6 7 8 9 100.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00F / Fo

Design Cycle

Figure 22. Reduction of cost functional with design cycles for Gd10 sub300.

46


X

Y

Z

Figure 23. A view of designed hull Gd10 sub300.

47


x

z

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

Figure 24. Projection on the xz-plane of designed hull Gd10 sub300.

x

y

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-0.1

-0.05

0

0.05

0.1

0.15

Figure 25. Projection on the xy-plane of designed hull Gd10 sub300.

48


y

z

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

Figure 26. Bow view of designed hull Gd10 sub300.

49


y

z

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

Figure 27. Stern view of designed hull Gd10 sub300.

50


Fr

Rw,R

p

0.15 0.2 0.25 0.3 0.35 0.4 0.450

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

Go

Gd10

Rp

Rw

Rw

Rp

Figure 28. Wave resistance coefficient for base-line hull G0 and designed hull Gd10

sub300 (Rp by pressure integration and Rw by wave cut method).

51


X

Y

-1 0 1-1

-0.5

0

0.5

1

0.002

-0.002

0.002-0.002

0.002

0.002

Gd10

Go

Figure 29. Wave pattern of base-line hull G0 (lower half) and designed hull Gd10

sub300 (upper half).

52


x

η

-1.5 -1 -0.5 0 0.5 1-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Gd10Go

Figure 30. Wave elevation in the plane of symmetry of base-line hull G0 anddesigned hull Gd10 sub300.

53


X Y

Z

0.03

-0.006

-0.006

0.0 -0.006

Gd10Go

0.00.0

0.03

-0.008

Figure 31. Pressure contours (0.5F 2nCp) on bow of base-line hull G0 (left half)

and designed hull Gd10 sub300 (right half).

54


X Y

Z

-0.008

-0.002

0.002

-0.006

Gd10Go

0.0

0.0

-0.018

Figure 32. Pressure contours (0.5F 2nCp) on stern of base-line hull G0 (left half)

and designed hull Gd10 sub300 (right half).

55


x

df(x

)

-0.5 -0.25 0 0.25 0.5-4E-05

-2E-05

0

2E-05

4E-05

6E-05

8E-05

BowStern

Go

Gd10

Figure 33. Contribution of station strips (between x and x + dx) to wave resis-tance of base-line hull G0 and designed hull Gd10 sub300.

x

f(x)

-0.5 -0.25 0 0.25 0.5-1.0E-04

0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-04

5.0E-04

6.0E-04

BowStern

Go

Gd10

Figure 34. Contribution of all strips between bow (x = 0.5) and station x towave resistance of base-line hull G0 and designed hull Gd10 sub300.

56


-0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14y

-0.28

-0.26

-0.24

-0.22

-0.2

-0.18

-0.16

-0.14

-0.12

z

Stations at -0.5 <= x <=0 Stations at 0<=x<=0.5

-0.5 -0.25 0 0.25 0.5x

-0.25

-0.2

-0.15

-0.1

-0.05

0

z

Figure 35. Designed body Gd sub450 for optimal solution to the inverse problem;free surface z = 0, Froude number Fn = 0.450.

57


-0.5 -0.25 0 0.25 0.5x

-0.5

-0.25

0

0.25

0.5

0.75

1

Cp

Gd at Fr=0.450Go at Fr=0 (DB)Go at Fr=0.450

Figure 36. Pressure coefficient distributions on the top line of symmetry of de-signed body Gd sub450 and initial body G0.

58


Surface Ships

In this section, we demonstrate the use of the adjoint formulation for shape optimizationof surface ships. We seek hull forms of minimum wave resistance subject to certain geometricconstraints that will be given later. For the sake of comparison, we perform the optimizationusing a finite-difference for the gradient of the cost functional at all design cycles. We alsosolve the inverse problem in which a target pressure distribution on hull is specified and ahull form is sought. All results are based on double-body linearization of the free-surfaceboundary conditions.

Base-Line Ship G0

A base-line ship, which is denoted by G0, is constructed by a composite B-spline surfacepatches. Uniform bi-cubic patches are used. The control points are uniformly spaced in the x-and z-directions and their offsets are placed on a mathematically defined surface. However,this surface is used only for the definition of G0 and is not needed for the optimizationprocedure. The plane y = 0 is a plane of symmetry and half the hull surface is parameterizedby (m + 1)(n + 1) control points. Each control point is identified by two indices (i, j) wherei = 0, 1, . . . , m and j = 0, 1, . . . , n. For later use, the two indices are combined into a onedimensional index k = j(m + 1) + i + 1. The streamwise positions xij of the control pointsvary only with index j and their depths zij vary only with index i. There are (m− 2)(n− 2)surface patches and each patch is subdivided into a small number of panels (mp × np). Thetotal number of panels on half the hull is mpnp(m − 2)(n − 2). Grid sensitivity studies canbe easily achieved by varying one or more of the four parameters in this expression.

For m = 7 and n = 11, the B-spline control points indices are depicted in Figure 37. Thefirst station of bow corresponds to the line j = 1, i = 1, . . . , 6, and the last station of sternto the line j = 10, i = 1, . . . , 6. The keel corresponds to the line i = 1, j = 1, . . . , 10, and thewaterline to the line i = 6, j = 1, . . . , 10. Each patch is subdivided into (mp × np) = (5 × 8)panels which gives 73 stations and 26 waterlines and a total of 1800 panels on half the hull.Different projections of G0 are depicted in Figures 38–40.

The ship length is used as a reference length. The first station of bow is at x = 0.5and the last station of stern is at x = −0.5. The hull is symmetric fore and aft of the mid-ship section (x = 0). Some geometric characteristics of G0 are: waterline beam B = 0.116(L/B = 8.62), draft T = 0.0544 (B/T = 2.13) which is uniform along the ship length,displacement V0 = 0.00396, wetted surface area S0 = 0.168, and block coefficient CB = 0.628.

The flow around G0 is solved using SWAN-v2.2 code. The basis flow around which the freesurface conditions are linearized is the double-body flow (similar to Dawson’s linearization,see Nakos and Sclavounos, 1990, for more details). The wave resistance coefficient as afunction of Froude number (based on ship length) is shown in Figure 41. There are twocurves corresponding to two different methods of calculating wave resistance in SWAN-v2.2code. The curve denoted by Rw is based on transverse wave cut method while Rp is obtainedby integrating surface pressure including a waterline contribution.

59



The wave resistance of G0 has a local maximum near Fn = 0.287 due to constructivewave interference. We choose to optimize the hull at this Froude number subject to geo-metric constraints. The objective functional is the wave resistance given by Equation (6).The design variables are the offsets of the B-spline control points while their depths andstreamwise locations are held fixed. However, not all of the control points are available asdesign variables. This is because the first station, the last station, and the keel must all havezero offsets. The control points which are free to change correspond to indices i = 2, . . . , mand j = 2, . . . , (n − 2). Therefore, the number of design variables is ndv = (m − 1)(n − 3),which is 48 in the present example.

Since the depths and streamwise positions of the control points are fixed, the optimizedhull will have a profile in the plane of symmetry as that of the original hull G0. Twoconstraints are imposed on the optimized hull: its displacement Vd satisfies an equalityconstraint Vd = V0, whereas its wetted surface area Sd satisfies an inequality constraintSd < 1.3S0. This means that the displacement of the optimized hull is equal to that of theoriginal hull but its surface may be larger than the original surface.

We must also specify side constraints. To this end, we let y0k denote the initial values of

the design variables that give the original hull G0 and let d1 and d2 denote the minimumand maximum of these values, respectively. In terms of these quantities, we define lowerand upper limits for each design variable: yl

k = max(d1, 0.5y0k) and yu

k = min(1.2d2, 1.5y0k).

Finally, we specify side constraints by: ylk < yk < yu

k . This constrained optimization problemis solved using a generalized reduced gradient method (GRG) as detailed by Belegundu andChandrupatla (1999, page 176). In this method, inequality constraints are replaced withequality constrained by introducing slack variables.

It is important to establish credibility of the adjoint method for computing the gradientof the cost functional with respect to the design variables yk. For this purpose, comparisonis made between the gradient found by the adjoint and that by a finite-difference method. Inthe latter method, the control variables yk are perturbed one at a time and the cost functionalis computed on each perturbed hull. With the cost function on the unperturbed hull known,the gradient is computed by a first-order forward finite-difference method. There are 48control variables, and thus the flow field has to be recalculated on 48 different perturbedhulls in addition to the unperturbed one. This should be compared with only two solutionsfor the adjoint approach; one for the velocity potential and the other for the adjoint function.The 48 components of the gradient computed by the two methods are depicted in Figure 42against the index k which is defined by Figure 37. The overall agreement between the twomethods is satisfactory. The discrepancies in the components of the two gradients afternormalization by the magnitude of the gradient vector are shown in Figure 43. The error isless than 6 percent.

It is instructive to investigate the contribution of the adjoint term to the gradient of thecost functional. In Equation (80) we substitute ψ = 0 and determine the gradient using the

60


remaining terms, which is equivalent to setting δφ = 0 in Equation (13). The gradient andthe corresponding error of this (incorrect) method of computing the gradient are shown inFigures 44 and 45. Neglecting the contribution of the adjoint results in a 30 percent errorand gives a qualitatively incorrect gradient. Figure 44 clearly shows that the present adjointformulation is correct. The discrepancy between finite-difference method and the adjointmay be reduced by improving the waterline conditions or considerations of the adjoint forthe double-body flow.

The reduction of the cost functional Equation (6) obtained by GRG method is shown inFigure 46. The iteration process is terminated after 20 design cycles since the reduction isdeemed sufficient while further reduction becomes very slow in the current GRG method.The designed hull, which is denoted by Gd20 S287, is shown in Figures 47–50. We note thatthe bow and stern regions are pinched while a small bulge develops at the mid ship region.The stern region suffers the most modifications. The maximum waterline beam is 0.133 andthe wetted surface area is 0.173; which is 3.0 percent larger than that of G0.

The wave resistance coefficient of Gd20 S287 over a range of Froude number is comparedwith that of G0 in Figure 51. At the optimization Froude number Fn = 0.287, the coefficientRw for G0 is 2.70 × 10−3 and for Gd20 S287 it is 0.204 × 10−3. The reduction in the waveresistance is approximately 92 percent. It is important to recall that this result is based onpotential flow theory. We also note that the wave resistance of the designed hull is lowerthan that of G0 over the Froude number range 0.25 to 0.35, outside this range the trend isreversed.

The lower wave resistance of Gd20 S287 manifests itself in the wave pattern and water-linewave elevation shown in Figures 52 and 53, respectively. As evident in Figure 53, the sternwave is significantly reduced while the bow wave is slightly reduced. It is also interestingto note that there are two waves on the base-line hull G0 whereas only one wave can beidentified on the designed hull Gd20 S287. The pressure distribution on bow is shown inFigure 54 and on stern in Figure 55. The left side of each figure is for G0 and the rightside for Gd20 S287. The major changes in the pressure distribution take place in the sternregion where a strong suction on G0 is practically removed and the pressure becomes nearlyuniform at free stream value, Cp = 0. The removal of the low pressure region on the sternis the source of drag reduction on the designed hull. This conclusion is also supported byFigures 56 and 57. The contributions of small strips between stations x and x + dx to thecost functional Equation (6) is shown as a function of x in Figure 56. The cumulative effectof all stations between bow and station x is shown in Figure 57. Almost all of the dragreduction on the designed hull Gd20 S287 is due to the improved pressure distribution on thelast 25 percent of the hull. This is a bit unfortunate because viscous effects and interactionswith propeller/jet dominate the flow field in this region. Considerations of these effects andinteractions are necessary for the design of the stern region.

61


Optimization Using Finite Difference for Gradient Computation

Using finite-difference for gradient calculations is very expensive, especially for 48 designvariables, but it can be used occasionally to verify a design obtained by the adjoint formula-tion. In the previous section, comparison is made between the two methods for computingthe gradient on G0 with 48 design variables and 7800 panels (1800 panels on half the hull).In this section, the optimization problem is solved using a finite-difference method for gra-dient calculations throughout 20 design cycles. The same 48 control points of the previoussection are used as design variables but the number of panels per patch is reduced so thatthe computation can be completed in a reasonable time. Here, the total number of panelsis reduced to 3600 (900 on half the hull) and the design takes 200 hours of “wall” time onSGI-R10000. The adjoint formulation takes 14 hours to complete 20 design cycles on thiscoarse grid. All adjoint results shown here have been obtained on the fine resolution of 7800panels, and it takes 48 hours to complete 20 design cycles. A finite-difference design wouldhave taken an estimated 700 hours.

At the end of 20 design cycles of the finite-difference design, the 48 offsets are used togenerate a refined grid with a total of 7800 panels (1800 on half the hull). The form of thishull, its wave pattern and resistance are now compared with those obtained by the adjointmethod on the fine grid. The bow and stern are compared in Figures 58 and 59, respectively.There are discernible differences between the two hulls, especially in the bow region. This isdue the differences in gradient vectors of the two methods, which are compounded throughthe 20 design cycles. Nonetheless, the hydrodynamic performances of these two hull formsare remarkably similar. This is shown by the wave patterns and waterline elevations for thetwo hulls depicted in Figures 60 and 61, respectively. Moreover, the force distributions onthe two hull forms shown in Figure 62 are in close agreement. Finally, the wave resistancecoefficients over a range of Froude number are compared in Figure 63. Taken collectively,these results affirm that the two hull forms have the same hydrodynamic performance. Thisseemingly fortuitous agreement in the hydrodynamic performance can be explained by thefact that the two hull forms have almost identical sectional area distributions as shown inFigure 64. The waterline of the two hulls are also shown in Figure 65 which shows a smalldifference at mid-ship.

A Target Pressure Distribution at Froude Number Fn = 0.287

This section concerns the application of the present shape optimization method to thesolution of an inverse problem in ship hydrodynamics. The objective is to design a hullform so that the pressure distribution on its surface matches a target distribution to bespecified by the designer. Experience in ship hydrodynamics plays an important role in thespecification of such a distribution. For example, the pressure distribution on an existinghull is first determined, the designer may choose to introduce modifications as to mitigateflow separation or cavitation in a certain locality. An exact solution to the inverse problemmay not exist. The use of shape optimization techniques provides a hull form whose pressure

62


distribution is as nearly as possible to the target distribution. The cost functional is givenby Equation (9).

To demonstrate the method, the target pressure distribution is given by the double-body pressure coefficient distribution on the base-line hull G0. This is an interesting targetpressure because it gives zero wave drag on G0. Although Froude number (or gravitational)effects do not play a role in the determination of the double-body pressure coefficient, theoptimized hull form depends on the Froude number at which optimization is performed.Moreover, the wave drag of the designed hull may not be zero. Even though the pressurecoefficient at each panel center of the optimized hull matches the target distribution, thearea and orientation of each panel may have changed from those of the corresponding panelof G0. The double-body pressure distribution gives zero drag only on G0 but not necessarilyon a deformed shape that follows from it.

The inverse problem is solved at Froude number Fn = 0.287. The constraints on dis-placement and wetted surface area as well as side constraints are exactly the same as givenon pages 60–61 for wave resistance minimization at Froude number Fn = 0.287. However,because a different objective functional is specified, it is important to establish credibility ofthe adjoint method for computing the gradient. Here again comparison is made between thegradient found by the adjoint and that by a finite-difference method. The 48 componentsof the gradient computed by the two methods are depicted in Figure 66 against the indexk which is defined by Figure 37. The overall agreement between the two methods is verygood. In Figure 67 we depict the difference between the two methods after normalization bythe magnitude of the gradient vector. The error is less than 5 percent.

The reduction of the cost functional Equation (9) is shown in Figure 68. The iterationprocess is terminated after 10 design cycles since the reduction is deemed sufficient, in factthe cost functional levels off after 7 cycles.

The target and actual pressure distributions on the designed hull Gd10 are shown inFigure 69 as a function of x at four different depths. The first plot at kz = 1 correspondsto the row of panels closest to the free surface, and the fourth plot at kz = 25 correspondsto the last row of panels at the keel. The pressure distribution on the base-line hull G0

at Froude number Fn = 0.287 is given by the dashed line. This pressure distribution issignificantly different from the double-body pressure distribution on the same hull (shownby the open circles) and which is used as a target pressure. It is evident in these figures thatthe pressure on the optimized hull is close to the target pressure almost everywhere on hullexcept in the bow region (x = 0.5) near the waterline (kz = 1). This is expected becauseshape deformation downstream of bow has little influence on bow flow due to the hyperboliccharacter of the free-surface condition.

The designed hull, which is denoted by Gd10 S287P, is shown in Figures 70–73. It issurprising that the bow region is strongly pinched but the stern suffers less modifications. Themaximum waterline beam is 0.127 and the wetted surface area is 0.172; which is 2.3 percentlarger than that of G0. The body plan depicted in Figure 72 shows that the hull lacks

63


symmetry fore and aft of the mid ship section.

The optimization iteration is continued for another 20 cycles but no changes are observedin hull form. The wave resistance coefficient over a range of Froude number is shown inFigure 74. At the design Froude number Fn = 0.287, the wave drag coefficient is Rw =0.907 × 10−3, thus the drag reduction is 65 percent of the base-line hull. Although theobjective functional in this design is for a target pressure, the wave resistance of the designedhull shows favorable reduction over a relatively high Froude numbers up to 0.4.

Weakly Wall-Sided Design at Froude Number Fn = 0.287

The effects of imposing more geometric constraints on the ship previously optimized tominimize wave resistance at Fn = 0.287 (pages 60–61) is investigated in this and the nextsection. This task is made easy by the flexibility of the B-spline parameterization. In thissection, the hull surface is constrained to be normal to the undisturbed free surface z = 0,that is to say ∂y/∂z = 0 at z = 0. With reference to Figure 37, this condition is obtained ifthe last three control points (i = 5, 6, 7) at any column, j, have the same value. Of coursethis will reduce the number of design variables from 48 to 32. All other geometric and sideconstraints of the previous optimization apply here. The optimized hull is called “weaklywall-sided” design.

The designed hull, which is denoted by Gd20 S287WS, is shown in Figures 75 and 76. Itis clear that the zero-slope condition is satisfied on the free surface z = 0. The geometriccharacteristics of this hull are similar to those of Gd20 S287; hence they have similar hydro-dynamic performances. For instance the water-line elevations are shown in Figure 77; thewall-sided hull shows smoother and lower wave at mid ship.

Partially Constrained Stern at Froude Number Fn = 0.287

In this section, we reconsider the optimization of the base-line hull G0 and use a subsetof the control points used for the previous wave resistance minimization on pages 60–61.That previous, optimized hull shows major deformation in the stern region and less in thebow region. Therefore, we partially constrain the stern region by excluding the last tworows of control points, those corresponding to j = 8 and 9 in Figure 37. Thus the numberof control points is reduced to 36. All geometric constraints of ship S287 remain effective inthis section.

At the end of 20 design cycle, the optimized hull, which is denoted by Gd20 S287B, is shownin Figures 78–80. A small bulge develops on the bow about half depth, but no significantdeformation appears on the stern. At the optimization Froude number Fn = 0.287, the waveresistance coefficient of Gd20 S287B is Rw = 0.358 × 10−3, which shows a drag reduction of86 percent.

The wave pattern and water-line wave elevation for the two ships Gd20 S287 and Gd20

S287B are compared in Figures 81–82, respectively. As evident in Figure 82, bow wave isreduced but the reduction of stern wave is less than with ship S287 for which the stern

64


region is not constrained. Finally, the force on hull between bow and station x is shown inFigure 83. The bow region (x < 0.25) shows reduction of force due improved flow on bow,but soon that reduction is lost on mid-ship. The stern region also shows less reduction inforce.

65


0 1 2 3 4 5 6 7 8 9 10 110

1

2

3

4

5

6

7

j

i

k= 24 32 40 48 56 64 72 80

19 27 35 43 51 59 67 75

23

22

21

20

79

78

77

76

Bow Stern

Keel

WL

Figure 37. Indices of B-spline control points for base-line surface ship G0.

66


X Y

Z

Figure 38. Base-line hull G0.

67


x

y

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-0.1

-0.05

0

0.05

0.1

Figure 39. Projection on the xy-plane of base-line hull G0.

y

z

-0.05 0 0.05-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

Figure 40. Body plan of base-line hull G0.

68


Fr

Rw,R

p

0.15 0.20 0.25 0.30 0.35 0.40 0.450.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040

0.0045

0.0050

0.0055

0.0060

Rp

Rw

Go

Figure 41. Wave resistance coefficient for base-line hull G0 (Rp by pressure inte-gration and Rw by wave cut method).

69


k16 24 32 40 48 56 64 72 80 88

-0.0015

-0.0010

-0.0005

0.0000

0.0005

0.0010

0.0015

0.0020

Gradient


Figure 42. Gradient of cost function computed by finite-difference and adjointmethods, hull G0, Fn = 0.287.

k

erro

r

16 24 32 40 48 56 64 72 80 88-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05


70


k16 24 32 40 48 56 64 72 80 88

-0.0015

-0.0010

-0.0005

0.0000

0.0005

0.0010

0.0015

0.0020

Gradient

AdjointFinite Differenceδφ = 0

Figure 44. Gradient computed with and without contribution of adjoint (ψ = 0)and finite-difference method, Fn = 0.287.

k

erro

r

16 24 32 40 48 56 64 72 80 88

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

Adjointδφ = 0


71


Design Cycle0 2 4 6 8 10 12 14 16 18 20

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00F / Fo

Figure 46. Reduction of cost functional with design cycles for Gd20 S287.

72


X Y

Z

Figure 47. Water-line view of designed hull Gd20 S287.

73


Y

Z

X

Figure 48. Keel view of designed hull Gd20 S287.

74


x

y

-0.5 -0.25 0 0.25 0.5

-0.1

-0.05

0

0.05

0.1

0.15

Figure 49. Projection on the xy-plane of designed hull Gd20 S287.

y

z

-0.05 0 0.05-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

Figure 50. Body plan of designed hull Gd20 S287.

75


Fr

Rw,R

p

0.15 0.2 0.25 0.3 0.35 0.4 0.450.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040

0.0045

0.0050

0.0055

0.0060

Rp

Rw

Rp

Rw

Go

Gd20


S287 (Rp by pressure integration and Rw by wave cut method).

76


X

Y

-1.5 -1 -0.5 0 0.5-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

0.001-0.001

0.001-0.001 Go

Gd20

Figure 52. Wave pattern on base-line hull G0 (lower half) and designed hull Gd20

S287 (upper half).

77


x

η

-1.5 -1 -0.5 0 0.5 1-0.015

-0.01

-0.005

0

0.005

0.01

0.015

Gd20

Go

Figure 53. Water-line wave elevation on base-line hull G0 and designed hull Gd20 S287.

78


Y

Z

-0.05 0 0.05-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01 0.0090.0

-0.007-0.007

Go Gd20

0.008 0.0

Figure 54. Pressure contours (0.5F 2nCp) on bow of base-line hull G0 (left half)

and designed hull Gd20 S287 (right half).

79


Y

Z

-0.05 0 0.05-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01 0.0-0.01

-0.007-0.008

Go Gd20

0.004 0.00.005

Figure 55. Pressure contours (0.5F 2nCp) on stern of base-line hull G0 (left half)

and designed hull (Gd20 S287 (right half).

80


x

df(x

)

-0.5 -0.25 0 0.25 0.5-1.5E-05

-1E-05

-5E-06

0

5E-06

1E-05

1.5E-05

BowStern

Go

Gd20

Figure 56. Contribution of station strips (between x and x + dx) to wave resis-tance of base-line hull G0 and designed hull Gd20 S287.

81


x

f(x)

-0.5 -0.25 0 0.25 0.5-5.0E-05

0.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

BowStern

Go

Gd20

Figure 57. Contribution of all strips between bow x = 0.5 and station x to waveresistance of base-line hull G0 and designed hull Gd20 S287.

82


y

z

-0.05 0 0.05-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

Finite Difference AdjointBow

Figure 58. Comparison of hull forms by adjoint (right half) and finite difference(left half), bow view.

83


y

z

-0.05 0 0.05-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

Finite Difference AdjointStern

Figure 59. Comparison of hull forms by adjoint (right half) and finite difference(left half), stern view.

84


X

Y

-1.5 -1 -0.5 0 0.5-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1Adjoint

Finite Difference

0.001-0.001

0.001-0.001

Figure 60. Comparison of wave pattern on finite-difference hull (lower half) andadjoint hull (upper half).

85


x

η

-1.5 -1 -0.5 0 0.5 1-0.015

-0.01

-0.005

0

0.005

0.01

0.015


Figure 61. Comparison of water-line wave elevation on finite-difference hull andadjoint hull.

86


x

f(x)

-0.5 -0.25 0 0.25 0.5-5.0E-05

0.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04Gd20 AdjointGd20 Finite DifferenceGo

BowStern

Figure 62. Contribution of all strips between bow x = 0.5 and station x to waveresistance of finite-difference hull, adjoint hull, and base-line hull.

87


Fr

Rw

0.15 0.2 0.25 0.3 0.35 0.4 0.450.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040

0.0045

0.0050

0.0055

0.0060

Gd20 Finite DifferenceGd20 AdjointDesign Froude Number

Figure 63. Wave resistance coefficient for finite-difference hull and adjoint hull(Rw by wave cut method).

88


x

A

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

Gd20 AdjointGd20 Finite DifferenceGo

Figure 64. Sectional area distribution for finite-difference hull and adjoint hull.

89


x

y WL

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Gd20 AdjointGd20 Finite DifferenceGo

Figure 65. Waterline for finite-difference hull and adjoint hull.

90


k16 24 32 40 48 56 64 72 80 88

-0.0015

-0.0010

-0.0005

0.0000

0.0005

0.0010

0.0015

Gradient


Figure 66. Gradient computed by finite-difference and adjoint methods, hull G0,Fn = 0.287.

k

erro

r

16 24 32 40 48 56 64 72 80 88

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

Figure 67. Error in gradient components normalized by magnitude of gradientvector, hull G0, Fn = 0.287.

91


0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1F / Fo

Design Cycle

Figure 68. Reduction of cost functional for hull Gd10 S287P.

92


x

P,P

d,P

o

-0.5 -0.25 0 0.25 0.5

-0.1

-0.05

0

0.05

0.1

0.15kz=8

Po

P

Pd

xP

,Pd,

Po

-0.5 -0.25 0 0.25

-0.1

-0.05

0

0.05

0.1

0.15kz=16

Po

PPd

x

P,P

d,P

o

-0.5 -0.25 0 0.25 0.5

-0.1

-0.05

0

0.05

0.1

0.15kz=1

Po

P

Pd

x

P,P

d,P

o

-0.5 -0.25 0 0.25

-0.1

-0.05

0

0.05

0.1

0.15

kz=25

PPo

Figure 69. Pressure distribution on hull, P =pressure on designed hull Gd10 S287P, Pd = target pressure, andP0 = pressure on G0, Fn = 0.287.

93


X Y

Z

Figure 70. Water-line view of designed hull Gd10 S287P.

94


Y

Z

X

Figure 71. Keel view of designed hull Gd10 S287P.

95


x

y

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-0.1

-0.05

0

0.05

0.1

0.15

Figure 72. Projection on the xy-plane of designed hull Gd10 S287P.

y

z

-0.05 0 0.05-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

Figure 73. Body plan of designed hull Gd10 S287P.

96


Fr

Rw,R

p

0.15 0.2 0.25 0.3 0.35 0.4 0.450.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040

0.0045

0.0050

0.0055

0.0060

Rp

Rw

Gd20

Rp

Rw

Go


S287P (Rp by pressure integration and Rw by wave cut method).

97


x

y

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-0.1

-0.05

0

0.05

0.1

0.15

Figure 75. Projection on the xy-plane of designed hull Gd20 S287WS.

y

z

-0.05 0 0.05-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

Figure 76. Body plan of designed hull Gd20 S287WS.

98


x

η

-1.5 -1 -0.5 0 0.5 1-0.015

-0.01

-0.005

0

0.005

0.01

0.015

S287-Wall-SidedS287

Figure 77. Water-line wave elevation on hull Gd20 S287 and weakly wall-sidedhull Gd20 S287WS.

99


X Y

Z

Figure 78. Water-line view of designed hull Gd20 S287B.

100


x

y

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-0.1

-0.05

0

0.05

0.1

0.15

Figure 79. Projection on the xy-plane of designed hull Gd20 S287B.

y

z

-0.05 0 0.05-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

Figure 80. Body plan of designed hull Gd20 S287B.

101


X

Y

-1.5 -1 -0.5 0 0.5-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

0.001-0.001

0.001-0.001

Gd20-S287B

Gd20-S287

Figure 81. Comparison of wave pattern on hull Gd20 S287 (lower half) and par-tially constrained stern hull Gd20 S287B (upper half).

102


x

η

-1.5 -1 -0.5 0 0.5 1-0.015

-0.01

-0.005

0

0.005

0.01

0.015

S287BS287

Figure 82. Water-line wave elevation on hull Gd20 S287 and partially constrainedstern hull Gd20 S287B.

103


x

f(x)

-0.5 -0.25 0 0.25 0.5-5.0E-05

0.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

Gd20-S287BGd20-S287Go

Figure 83. Contribution of all strips between bow x = 0.5 and station x to waveresistance of base-line hull G0, designed hull Gd20 S287, and partiallyconstrained stern hull Gd20 S287B.

104


Conclusions and Recommendations

An efficient gradient-based optimization method has been applied successfully to thehydrodynamic design of submarine configurations operating near a free surface and surfaceships. The efficiency of the method stems from the use of an adjoint formulation for thegradient of the cost functional with respect to design variables. The flow model is given by theclassical potential flow theory with Kelvin’s or Dawson’s linearized free-surface condition.Submarine shapes are optimized for minimum wave resistance subject to constraints ondepth, displacement and surface area. For surface ships, a base-line hull is first defined. Thishull is optimized for minimum wave resistance or for a target pressure distribution subjectto constraints. These constrained problems are solved by a generalized reduced gradientmethod. The inverse problem, in which a target pressure distribution on hull is specified,has been also presented. B-spline surfaces are used for geometry parameterizations and proveto be versatile for shape optimization.

In several examples, the accuracy of the adjoint formulation for free surface flows hasbeen validated by comparisons with direct calculations using a finite-difference method. Inall examples considered, the discrepancies between the two methods is less than 6 percentof the gradient vector magnitude. Furthermore, the geometric characteristics and hydrody-namic performance of an optimized hull using a finite-difference method for gradient havebeen compared with that using the adjoint approach. Although there are differences in thegeometric details of the two surface ship hull forms, they show almost identical wave pat-terns and wave resistance not only at the optimization Froude number Fn = 0.287 but alsoover a wide range of Froude numbers. This is because the two hull forms have nearly equalsectional area distributions.

A base-line hull which is a body of revolution typical of submarine configurations wasfirst analyzed. Its wave resistance was then minimized at the Froude number correspondingto a local maxima in the Fn-Resistance curve. The optimized hull forms are consistent withthe concept of destructive wave interference. These forms are strongly dependent on Froudenumber. A practically waveless hull has been obtained at Froude number of 0.230, andsignificant reductions in wave resistance have been obtained at other Froude numbers.

The use of B-spline parameterization has proven very useful in controlling local featuresof the designed hull forms. For example, a nearly wall-sided hull can be designed. Also, ahull form is optimized while its stern region is partially constrained.

The current generalized-reduced-gradient methods takes about 20, or less, design cycles toreach a minimum. All of the adjoint results presented in this chapter have been obtained ona fine grid of 7800 panels (1800 panels on half the hull). Computations have been performedon a single-processor SGI-R10000 workstation. Typical “wall” times to complete 20 designcycles are 48–72 hours. A finite-difference method would take an estimated 700–1000 hourson the same grid and for the same number of design variables.

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Recommendations for Future Work

The shape optimization technique presented in this report is very encouraging. It can beenhanced and applied to the design of more practical configurations. The following extensionsand applications are recommended:

1. Apply method to bulbous bows thereby relieving the modifications in the stern region.

2. Investigate the advantages and potential of cost functionals defined on the free surfaceor the downstream boundary.

3. Improve treatment of the adjoint water-line boundary conditions, and include the ef-fects of double-body flow adjoint on the free-surface condition.

4. Extend adjoint formulation to nonlinear free-surface flow.

5. Apply nonlinear method to the optimization of ships with transom sterns.

6. Apply method to the optimization of multi-hull ships.

7. Apply method to the optimization of hydro-propulsion systems for fast ships propelledby water jets.

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References

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Eckert, E., and Sharma, S. D., Bugwulste fur langsame, vollige Schiffe. Jahrb. Schift-bautech. Ges. 64 , 1970, 129–158; Erort, 159–171.

Froude, W., “Experiments upon the effect produced on the wave-making resistance ofships by length of parallel middle body,” The Papers of William Froude, The Institution ofNaval Architects, London, 1955.

Hsiung, C. C., “Optimal shape forms for minimum wave resistance,” J. Ship Research,Vol. 25, No. 2, 1995, pp. 95–116.

Huang, C-H., Chiang, C-C., and Chou, S-K., “An inverse geometry design problem inoptimizing hull surfaces,” J. Ship Research, Vol. 42, No. 2, 1998, pp. 79–85.

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Kelvin, Sir William Thomson, “On deep-water ship waves,” Mathematical and PhysicalPapers, Vol. IV, Cambridge University Press, 1910, pp. 394–418.

Kelvin, Sir William Thomson, “On the waves produced by a single impulse in water ofany depth, or in a dispersive medium,” Proceedings of the Royal Society of London, Ser. A.,Vol. 42, 1887, pp. 80–85.

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Lowe, T. W., Bloor, M. I. G., and Wilson, M. J., “The automatic functional design ofhull surface geometry,” J. Ship Research, Vol. 38, No. 4, December 1994, pp. 319–328.

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Nakos, D., and Sclavounos, P., “Ship motions by a three-dimensional Rankine panelmethod,” 18th Naval Hydrodynamic Symposium, pp. 21–40, 1990.

Pashin, V. M., Bushkovsky, V. A., and Amromin, E. L., “Determination of three-dimensional body forms from given pressure distribution on their surfaces,” J. Ship Research,Vol. 40, No. 1, 1996, pp. 22–27.

Pironneau, O., Optimal Shape Design for Elliptic Systems, Springer-Verlag, New York,1984.

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Ragab, S. A., “Hydrodynamic design of three-dimensional non-lifting surfaces,” PaperNo. FEDSM97-3414. The 1997 ASME Fluids Engineering Division Summer Meeting, Van-couver, Canada, June 22–26, 1997.

Soemarwoto, B., “The variational method for aerodynamic optimization using the Navier-Stokes equations,” ICASE Report No. 97-71 (Also NASA/CR-97-206277), NASA LangleyResearch Center, Hampton, VA, 1997.

Stoker, J. J., Water Waves, Interscience, New York, 1957.

Tuck, E. O., “Ship-hydrodynamic free-surface problems without waves,” J. Ship Research,Vol. 35, No. 4, 1991, pp. 277–287.

Tulin, M., and Oshri, O., “Free surface flows without waves; applications to fast shipswith low-wave resistance,” 20th Symposium on Naval Hydrodynamics, National AcademyPress, Washington, D.C. 1996, pp. 157–169.

Vanderplaats, G. N., Numerical Optimization Techniques For Engineering Design, 2nded., Vanderplaats Research and Development, Inc., Colorado Spring, Co., 1998.

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Wyatt, D. C., and Chang, P. A., “Development and assessment of a total resistanceoptimized bow for the AE 36,” Marine Technology, Vol. 31, No. 2, 1994, pp. 149–160.

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