In order to design an optimal floating breakwater with a high performance in a wide range of frequencies, 2D and 3D analyses are performed in this study. The design starts with seeking an optimal 2D model shape. For this purpose, an optimization method called Genetic Algorithm (GA) combined with Boundary Element Method (BEM) is employed as the main calculation method. The accuracy of BEM analysis is confirmed using several relations such as HaskindNewman and energy conservation relations. Moreover, since the investigated model will be an asymmetric shape, an experiment using a manufactured asymmetric model is also conducted to confirm that the present analysis could treat asymmetric body case correctly. From the experiment, a favorable agreement with numerical results can be found for both fixed and free motions cases which strengthen our confidence on the 2D analysis correctness. However, because the optimal performance obtained in 2D analysis is expected to be different for some extent from real application, the performance of the corresponding model in 3D case is also analyzed. Higher order boundary element method (HOBEM) is employed for this purpose. 3D Wave effect and its effect to the floating breakwater performance are analyzed and discussed. For consideration of real model construction and installation, drift forces induced by waves are also computed. It is shown from this study that the combination of GA and BEM is effective in obtaining an optimal performance model. Moreover, by computing its the corresponding 3D model, it can also be shown that the 3D wave effect is small on motion amplitude while the wave elevation is found to be in 3D pattern even for a longer body length.
Design Optimization of a Floating Breakwater by Faisal MAHMUDDIN A dissertation submitted in partial fulfillment for the degree of Doctor of Engineering in the Graduate School of Engineering Department of Naval Architecture and Ocean Engineering Division of Global Architecture Osaka University August 2012 “All humans are dead except those who have knowledge And all those who have knowledge are asleep, except those who good deeds.” Imam Ash-Shaafi’ee OSAKA UNIVERSITY Abstract Graduate School of Engineering Department of Naval Architecture and Ocean Engineering Doctor of Engineering by Faisal MAHMUDDIN In order to design an optimal floating breakwater with a high performance in a wide range of frequencies, 2D and 3D analyses are performed in this study The design starts with seeking an optimal 2D model shape For this purpose, an optimization method called Genetic Algorithm (GA) combined with Boundary Element Method (BEM) is employed as the main calculation method The accuracy of BEM analysis is confirmed using several relations such as Haskind-Newman and energy conservation relations Moreover, since the investigated model will be an asymmetric shape, an experiment using a manufactured asymmetric model is also conducted to confirm that the present analysis could treat asymmetric body case correctly From the experiment, a favorable agreement with numerical results can be found for both fixed and free motions cases which strengthen our confidence on the 2D analysis correctness However, because the optimal performance obtained in 2D analysis is expected to be different for some extent from real application, the performance of the corresponding model in 3D case is also analyzed Higher order boundary element method (HOBEM) is employed for this purpose 3D Wave effect and its effect to the floating breakwater performance are analyzed and discussed For consideration of real model construction and installation, drift forces induced by waves are also computed It is shown from this study that the combination of GA and BEM is effective in obtaining an optimal performance model Moreover, by computing its the corresponding 3D model, it can also be shown that the 3D wave effect is small on motion amplitude while the wave elevation is found to be in 3D pattern even for a longer body length Acknowledgements I am sincerely and heartily grateful to my supervisor, Professor Masashi Kashiwagi, for his continous excellent guidance, care and patience throughout the course Despite his many other academic and professional commitments, he still could be able to provide me with an international top level atmosphere of research His abundant support and invaluable assistance that he gave truly help the progression and smoothness of my doctoral program It will be difficult to imagine having a better supervisor than him for my study My special thanks also to Professor Shigeru Naito for his constant caring and attention to my study Even though, we did not have much time for discussion but his encouragement is much indeed appreciated Besides them, I am also truly indebted and thankful to Professor Munehiko Minoura and Dr Guanghua He for their wise advice and insightful comments I also owe sincere and earnest thankfulness to Shimizu-san, for supporting me in the experiment Helping to remove obstacles and resolve problems have been crucial for achieving the experiment objectives Furthermore, I would like to say that it is a great pleasure to spend time with all of my very nice and friendly lab mates I highly appreciate the invitation to participate on sports activities and parties with halal food Thanks for the friendship and memories I would like to thank my family members, especially my mothers and sisters for the pray and encouragement to pursue this degree Finally, I would like to thank everybody who was important to accomplish the dissertation, as well as expressing my apology that I could not mention personally one by one Osaka, August 2012 Faisal Mahmuddin v Contents Abstract iv Acknowledgements v List of Figures ix List of Tables xi Abbreviations xiii Symbols xv Introduction 1.1 Background 1.2 Study Objectives and Organization Theory of 2D Optimization Method 2.1 Genetic Algorithm (GA) 2.1.1 Algorithm Principle 2.1.2 Encoding and Decoding 2.1.3 Genetic Operators 2.1.4 Shape Parameterization 2.1.5 Fitness Function 2.2 2D Boundary Element Method 2.2.1 Boundary Conditions 2.2.2 Boundary Integral Equation and Green Function 2.2.3 Hydrodynamics Forces 2.2.4 Equation of Motions 2.2.5 Reflection and Transmission Coefficient 2.2.6 Numerical Calculation of Velocity Potentials 1 5 10 12 13 13 16 19 22 27 28 Model Experiment 31 3.1 Introduction 31 vii Contents 3.2 3.3 3.4 3.5 3.6 Manufactured Model Experiment Preparation 2D Water Channel Experiment Setup Results and Analysis viii 32 33 34 35 36 Optimization Results Analysis 41 4.1 Parameters and Constraints 41 4.2 Results and Analysis 44 3D Performance Analysis 5.1 Solution Method 5.1.1 Mathematical Formulations 5.1.2 Higher-order Boundary Element Method (HOBEM) 5.1.3 Hydrodynamic Forces 5.1.4 Wave Elevation on the Free Surfaces 5.2 Computation Results and Discussion 51 52 52 56 58 66 67 Conclusions 79 Bibliography 81 List of Figures 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Workflow of GA Example of chromosomes and genes Body surface division Bezier Curve Definition of fitness 2D coordinate systems Coordinate system for an asymmetric floating body Coordinate system and notations of asymmetric body 10 10 12 13 16 22 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 Shape, notations and coordinate system of tested model Manufactured model used in the experiment Oscillation table Wave channel Experiment setting Transmission coefficent in fixed-motion case Motions amplitude and phase Transmission coefficent in free-motion case 31 33 34 35 35 37 38 39 4.1 4.8 The average and maximum values of fitness (𝑃 𝐼) in GA computation with 𝑃𝑚 =0 and 𝑃𝑐 =0.5 The average and maximum values of fitness (𝑃 𝐼) in GA computation for 𝑃𝑚 =0.5 and various values of 𝑃𝑐 𝑓𝑚𝑎𝑥 and 𝐿𝑊 𝐿 of simulation with additional criteria Fittest model and its performance in some particular generations Modified final shape for the model Transmission coefficients of the modified final model and corresponding rectangular shape Reflection and transmission coefficients of optimized model for fixedmotion case Body motion amplitudes of optimized 2D model 48 49 5.1 5.2 5.3 5.4 Coordinate system in the 3D analysis Quadrilateral 9-node Lagrangian element 3D model shape Body motion amplitudes of 3D model for 4.2 4.3 4.4 4.5 4.6 4.7 ix 𝐿/𝐵 = 42 43 44 46 47 47 52 57 68 69 List of Figures 3D Reflection (left) and transmission (right) wave coefficients for 𝐿/𝐵 = : (a) (b) for fixed motion case, (c) (d) for free motion case 5.6 Body motion amplitudes of 3D model for 𝐿/𝐵 = 5.7 3D Reflection (left) and transmission (right) wave coefficients for 𝐿/𝐵 = : (a) (b) for fixed motion case, (c) (d) for free motion case 5.8 Body motion amplitudes of 3D model for 𝐿/𝐵 = 20 5.9 3D Reflection (left) and transmission (right) wave coefficients for 𝐿/𝐵 = 20 : (a) (b) for fixed motion case, (c) (d) for free motion case 5.10 Bird’s-eye view of 3D wave field around a body of 𝐿/𝐵 = for wavelength of 𝜆/𝐵=3.0 and 6.0 5.11 Bird’s-eye view of 3D wave field around a body of 𝐿/𝐵 = 20 for wavelength of 𝜆/𝐵=3.0 and 6.0 5.12 Wave drift forces computed by 2D and 3D methods for a body of 𝐿/𝐵 = 20 for both cases of fixed and free motions x 5.5 70 71 72 73 74 75 77 78 Chapter 3D Performance Analysis 68 Figure 5.3: 3D model shape angle 𝛽 of regular incoming wave is set equal to 𝛽 = −90 deg so that the situation corresponds to the 2D case and the results for the body motions and the reflection and transmission wave coefficients can be compared with 2D results; thereby 3D effects on those quantities can be discussed Unlike 2D case, the wave amplitude in 3D results may vary depending on the location on the free surface Thus different positions along the 𝑦-axis (centerline of the body) are considered for the wave measurement The distance of these positions from the origin of the coordinate system is taken equal to 𝑦/𝑏 = 4, 10, and 18 for the reflection wave and 𝑦/𝑏 = −4, −10, and −18 for the transmission wave (Note that the incident-wave component is subtracted from Eq (5.73) in the definition of the reflection wave.) In order to investigate 3D effects depending on the longitudinal length of the body, we have computed for different body lengths; those are 𝐿/𝐵=2, 8, and 20 The hydrodymnamic forces are computed, but discussion in this study will be focused on the difference between 2D and 3D results in the amplitude of body motions and the reflection and transmission wave coefficients In numerical computations, only half of the body was discretized with the symmetry relation with respect to 𝑥 taken into account Then to keep sufficient accuracy, a larger number of panels was used, although the results of HOBEM are relatively very accurate Chapter 3D Performance Analysis 69 Specifically, the total number of panels used is 408 for 𝐿/𝐵 = 2, 638 for 𝐿/𝐵 = 8, and 1098 for 𝐿/𝐵 = 20 One panel consists of nodal points and thus the total number of unknowns was 1689, 2629, and 4509 for 𝐿/𝐵 =2, 8, and 20, respectively As already described, the numerical accuracy was checked through the HaskindNewman and energy-conservation relations and found to be very satisfactory with these panels and unknowns Computed results for a 3D body with 𝐿/𝐵 = are shown in Fig 5.4 for the amplitude of body motions and in Fig 5.5 for the reflection and transmission waves Figs 5.5 (a) and 5.5 (b) are for the diffraction problem and Figs 5.5 (c) and 5.5 (d) are for the case of all motions free 3D Motions Amplitude L/B= 3.5 Amplitude/ζa(.k) Sway Heave Roll 2.5 1.5 0.5 0 λ∞/B=π/Kb Figure 5.4: Body motion amplitudes of 3D model for 𝐿/𝐵 = Chapter 3D Performance Analysis 70 3D Transmission Coefficient L/B= 3D Reflection Coefficient L/B= 2 Fixed Motions Case Fixed Motions Case 1.5 y/b= y/b= 10 y/b= 18 Amplitude/ζa Amplitude/ζa 1.5 0.5 -y/b= -y/b= 10 -y/b= 18 0.5 λ∞/B=π/Kb (a) λ∞/B=π/Kb 3D Transmission Coefficient L/B= 3D Reflection Coefficient L/B= 2 Free Motions Case 1.5 Free Motions Case 1.5 y/b= y/b= 10 y/b= 18 Amplitude/ζa Amplitude/ζa (b) 0.5 -y/b= -y/b= 10 -y/b= 18 0.5 (c) λ∞/B=π/Kb 0 λ∞/B=π/Kb (d) Figure 5.5: 3D Reflection (left) and transmission (right) wave coefficients for 𝐿/𝐵 = : (a) (b) for fixed motion case, (c) (d) for free motion case From Fig 5.4 we can see that the body motions show very similar trend to the 2D results shown in Fig 4.8, but the amplitude particularly in heave is different On the other hand, the wave amplitudes shown in Fig 5.5 are very much different from those by the 2D analysis shown in Figs 4.6 and 4.7 Furthermore, the wave amplitudes in 3D results are dependent largely on the measurement position We can envisage that the incident wave is diffracted around the longitudinal tip side of the body and the wave field on the free surface is totally three dimensional It should be noted that regular fluctuation in the short wavelength region can be observed In order to resolve this fluctuation, computations have been performed at dense points of the wavelength with very small interval, and we found that Chapter 3D Performance Analysis 71 this fluctuation was caused by the so-called irregular frequencies As described in the numerical method, zero value of the velocity potential was specified on some interior free-surface points to get rid of the irregular frequencies However, the results show that this method is not effective for 3D problems Since computations are conducted at dense wavelengths in the present study, a mean line of this regular fluctuation may be considered as expected results and this fluctuation in the short wavelength region may be not a fatal problem in discussing 3D effects 3D Motions Amplitude L/B= 3.5 Amplitude/ζa(.k) Sway Heave Roll 2.5 1.5 0.5 0 λ∞/B=π/Kb Figure 5.6: Body motion amplitudes of 3D model for 𝐿/𝐵 = Computed results for a longer body of 𝐿/𝐵 = are shown in Figs 5.6 and 5.7 for the amplitudes of body motions and reflection and transmission waves, respectively Looking at the motion amplitudes in Fig 5.6 and comparing with Fig 4.8, we can see that all modes of motion become almost the same not only in the trend but also in the magnitude This implies that 3D effects on hydrodynamic forces are small enough if the length ratio of the body is taken up to 𝐿/𝐵 = However, the wave amplitudes are still different from the 2D results, although the global trend becomes similar For instance, for the case of fixed motions, the reflection wave is large and its coefficient is roughly equal to 1.0, and the transmission wave coefficient is smaller than 0.5 We can also see that, depending Chapter 3D Performance Analysis 72 3D Transmission Coefficient L/B= 3D Reflection Coefficient L/B= 2 Fixed Motions Case 1.5 Amplitude/ζa Amplitude/ζa 1.5 Fixed Motions Case 0.5 λ∞/B=π/Kb 0.5 y/b= y/b= 10 y/b= 18 -y/b= -y/b= 10 -y/b= 18 (a) λ∞/B=π/Kb (b) 3D Transmission Coefficient L/B= 3D Reflection Coefficient L/B= 2 1.5 1.5 Amplitude/ζa Amplitude/ζa Free Motions Case Free Motions Case 0.5 -y/b= -y/b= 10 -y/b= 18 0.5 y/b= y/b= 10 y/b= 18 0 (c) λ∞/B=π/Kb 0 λ∞/B=π/Kb (d) Figure 5.7: 3D Reflection (left) and transmission (right) wave coefficients for 𝐿/𝐵 = : (a) (b) for fixed motion case, (c) (d) for free motion case on the position and wavelength, the wave amplitude coefficient becomes larger than 1.0, which should be attributed to 3D effects in the free-surface wave In order to see whether more similar results to those in the 2D analysis would be obtained for a longer body, the body length was increased further to 𝐿/𝐵 = 20 Obtained results for the body motions and the reflection and transmission waves are shown in Figs 5.8 and 5.9, respectively The amplitudes of body motions are unchanged from the case of 𝐿/𝐵 = However, the results of wave amplitudes are still different but become similar further to the 2D results Chapter 3D Performance Analysis 73 3D Motions Amplitude L/B= 20 3.5 Amplitude/ζa(.k) Sway Heave Roll 2.5 1.5 0.5 0 λ∞/B=π/Kb Figure 5.8: Body motion amplitudes of 3D model for 𝐿/𝐵 = 20 Although the wave amplitude is still dependent on the position of measurement, the reflection wave coefficient fluctuates around 1.0 and decreases at wavelengths greater than 𝜆/𝐵 > 5.5 for the free-motion case, which is the same in trend as the 2D results Nevertheless, we can realize that 3D effects are large on the wave amplitude on the free surface even for a longer body of 𝐿/𝐵 = 20 3D Transmission Coefficient L/B= 20 3D Reflection Coefficient L/B= 20 2 Fixed Motions Case 1.5 Amplitude/ζa 1.5 Amplitude/ζa Fixed Motions Case y/b= y/b= 10 y/b= 18 0.5 -y/b= -y/b= 10 -y/b= 18 0.5 (a) λ∞/B=π/Kb 0 (b) λ∞/B=π/Kb Chapter 3D Performance Analysis 74 3D Transmission Coefficient L/B= 20 3D Reflection Coefficient L/B= 20 2 Free Motions Case Free Motions Case y/b= y/b= 10 y/b= 18 1.5 Amplitude/ζa Amplitude/ζa 1.5 0.5 -y/b= -y/b= 10 -y/b= 18 0.5 λ∞/B=π/Kb 0 (c) λ∞/B=π/Kb (d) Figure 5.9: 3D Reflection (left) and transmission (right) wave coefficients for 𝐿/𝐵 = 20 : (a) (b) for fixed motion case, (c) (d) for free motion case In order to observe the spatial variation of the free-surface wave around a floating breakwater, numerical computations for the bird’s-eye view of the wave field were performed for typical wavelengths; that is, 𝜆/𝐵 = 3.0 and 𝜆/𝐵 = 6.0 Computed results for a short-length body of 𝐿/𝐵 = are shown in Fig 5.10, where 5.10 (a) and 5.10 (b) are for 𝜆/𝐵 = 3.0 and 5.10 (c) and 5.10 (d) are for 𝜆/𝐵 = 6.0 Both cases of fixed and free motions are computed and shown These results are only for the real part ( i.e at time instant 𝑡 = 0) of the total wave elevation Therefore it may be difficult to distinguish the reflected and incident waves in the weather side, whereas in the lee side we can directly see the spatial distribution of transmitted wave and its correspondence to the results measured at selected points along the 𝑦−axis (which are shown in Fig 5.9 for the case of 𝐿/𝐵 = 2) Chapter 3D Performance Analysis 75 (a) (b) (c) (d) Figure 5.10: Bird’s-eye view of 3D wave field around a body of 𝐿/𝐵 = for wavelength of 𝜆/𝐵=3.0 and 6.0 Chapter 3D Performance Analysis 76 We can see from Fig 5.10 that the wave is relatively uniform for 𝜆/𝐵 = 6.0 but scattered by the body for 𝜆/𝐵 = 3.0 and the resulting wave pattern becomes three dimensional Computed results for a longer body of 𝐿/𝐵 = 20 are shown in Fig 5.11 Like above, 5.11(a) and 5.11(b) are for 𝜆/𝐵 = 3.0 and 5.11(c) and 5.11(d) are for 𝜆/𝐵 = 6.0, and both cases of fixed and free motions are shown to observe the effect of body motions (a) (b) (c) Chapter 3D Performance Analysis 77 (d) Figure 5.11: Bird’s-eye view of 3D wave field around a body of 𝐿/𝐵 = 20 for wavelength of 𝜆/𝐵=3.0 and 6.0 Looking at the wave in the lee side, we can confirm the correspondence to the results in Fig 5.9 measured at different points along the 𝑦-axis We can see that the effect of body motions is large in the wave pattern for both cases of 𝜆/𝐵 =3.0 and 6.0 In particular, at 𝜆/𝐵 = 6.0, the transmitted wave becomes large and really three dimensional, which is much different from the 2D results Finally computed results for the wave drift force are presented in Fig 5.12 as a comparison between 2D and 3D results Here the drift force is defined as positive when acting in the direction of incident-wave propagation The results in Fig 5.11 are just for a longer body of 𝐿/𝐵 = 20 , and we can see favorable agreement between 2D and 3D results except in a limited range of short wavelengths A discrepancy observed in this range might be attributed to insufficient accuracy in the integration with respect to 𝜃 in Eq (5.70) We can say from Fig 5.11 that the 2D analysis can be used for estimation of the wave drift force in the design Although the wave drift force and related mooring force are not considered in computing the wave-induced body motions in the present study, estimation of the wave drift force will be important in actual installation of a floating breakwater Chapter 3D Performance Analysis 78 Drift Forces (L/B= 20) 1.2 1 0.8 0.8 Fx Fx Drift Forces (L/B= 20) 1.2 0.6 0.4 0.4 Fixed Motions Case 2D Model 3D Model 0.2 (a) 0.6 Free Motions Case 2D Model 3D Model 0.2 λ∞/B= π/Kb 0 λ∞/B= π/Kb (b) Figure 5.12: Wave drift forces computed by 2D and 3D methods for a body of 𝐿/𝐵 = 20 for both cases of fixed and free motions Chapter Conclusions Using genetic algorithm (GA) and boundary element method (BEM) based on the potential-flow theory, a numerical analysis on the performance of floating breakwaters has been performed in both 2D and 3D cases Some important points found in this study are : a A numerical analysis using BEM on floating breakwater with asymmetric shape has been performed The accuracy and correctness of the analysis were confirmed using several relations and model experiment as well b A scheme based on GA combined with BEM has been exploited to find an optimal model of floating breakwater which has high performance in a wide range of frequencies c By computing for the corresponding 3D model of optimized shape, A difference performance from the 2D model was found However, the trend in variation with respect to the wavelength becomes similar for longer body which is known as 3D wave effect d 3D wave effects were not so large on the hydrodynamic forces and resultant wave-induced body motions 79 Chapter Conclusions 80 e The free-surface wave elevation was found to be spatially three dimensional even near the middle of a longer body f The drift forces for a longer body were almost the same in values as those for the 2D body Bibliographies [1] F Mahmuddin and M Kashiwagi, “Design optimization of a 2D asymmetric floating breakwater by genetic algorithm,” Proceeding of International Society of Offshore and Polar Engineers Conference (ISOPE), 2012 [2] M Kashiwagi and F Mahmuddin, “Numerical analysis of a 3D floating breakwater performance,” Proceeding of International Society of Offshore and Polar Engineers Conference (ISOPE), 2012 [3] M Kashiwagi, “Theory of floating body engineering,” Study Handout, pp 1– 79, 2006 [4] M Kashiwagi, H Yamada, M Yasunaga, and T Tsuji, “Development of a 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Floating Breakwater Seizando Shoten, Co Ltd, 2003 [12] T Haraguchi and S Ohmatsu, “On an improved solution of the oscillation problem on non-wall sided floating bodies and a new method for eliminating the irregular frequencies,” Trans of West Japan Society of Naval Architects, no 66, pp 9–23, 1993 [13] M Kashiwagi, “A calculation method for steady drift force and moment on multiple bodies (in japanese),” Bulletin Research Institute for Applied Mechanics, Kyushu University, no 170, pp 83–98, 1995 [14] H Maruo, “The drift force on a body floating in waves,” Journal of Ship Research, vol 4, no 3, pp 1–10, 1960 [15] J Newman, “The drift force and momoment on ships in waves,” Journal of Ship Research, vol 1, no 1, pp 51–60, 1967 ... pursue this degree Finally, I would like to thank everybody who was important to accomplish the dissertation, as well as expressing my apology that I could not mention personally one by one Osaka,... an undeterministic method, slightly different results might be obtained for different runs In this dissertation, the reflection and transmission coefficents, which are defined as the amount of incident