Drag Optimization of A Planing Vessel Based on the Stability Criteria Limits

One of the important phases of designing a craft is stability analysis and optimization of the drag force in cruise speed. In this research the longitudinal and lateral stability of a planing hull craft (DTMB 62 model 4667-1) is investigated with some semi-empirical formulae and the effects of some important design parameters are investigated on the limit of stability region. Also on the basis of these empirical formulations and by using a genetic algorithm the drag force is optimized in each constant cruise speed with the stability criteria limits at a constant beam or projected area. Aspect ratio, the longitudinal position of the gravity center and deadrise angle are the optimization parameters. The results show that the aspect ratio and the longitudinal position of the gravity center are two important parameters in optimizing the drag force and for this planing vessel the drag force can be reduced by 22%.


Introduction
Displacement and semi-displacement hulls increase the drag force in high speeds; therefore the planing hulls were designed at the beginning of the 20th century to reduce the drag force. In planing crafts, the hydrodynamic force is the dominant force and the weight of the vessel is supported mostly by hydrodynamic lift force (Tveitnes, 2001;Bertram, 2000). The aim of designing planing hulls is to achieve a high speed, but increasing the speed leads to the chances of more instability. Therefore from the first stage of designing the high speed hulls, investigation of the important parameters on both stability analysis and drag reduction was of interest. Experimental and numerical analysis can be used in these analyses, but for lowering the consumed time and cost, the semi-empirical formulations are among the best methods for optimization of drag force by considering stability criteria limits of the craft. Savitsky (1964), by using the experimental data of Day and Haag (1952), proposed some formulations to calculate the trim angle, wetted area and drag force. Also, he proposed a relation to calculate the limits of longitudinal stability trim angles region. Fridsma (1969) prepared a series of experimental work in rough water on prismatic planing hulls and investigated the effects of some design parameters on porpoising phenomena and drag force. Milton (1978) presented an analytical solution considering heave, pitch and surge equations of motion to analyze the porpoising instability of a prismatic hull, also Milton found the trim angle and Froude number of the starting point of the porpoising instability. Milton found that the effect of surge is negligible on the porpoising instability. Wellicome and Campbell (1984) investigated the transverse stability of a special kind of prismatic planing vessel. They found that one of the important parameters in the starting point of transverse instability is the vertical position of the center of gravity. They also studied the effect of some important parameters such as aspect ratio and deadrise angle on the starting point of transverse instability. Lewandowski (1997) prepared a set of experiments to predict the transverse stability of planing vessels and presented some semi-empirical formulae in terms of length, beam, deadrise angle, trim angle, draft and velocity of the vessel to calculate the maximum keel to center of gravity ratio as a stability limit of a vessel. Haddara and Jinsong (1997) used a neural network method to couple the heave and pitch motions of a vessel. Katayama (katayama, 2002) investigated the diving and transverse porpoising instability using some experimental data and relations for a high speed planing vessel. Mason et al. (2005) used genetic algorithms and artificial neural networks to optimize the resistance of a catamaran. Shen et al. (2013) compared the Savitsky method with a numerical method to calculate the resistance and trim angle of a catamaran in the speed range of 10 to 40 m/s and showed that the results of the Savitsky method were more accurate than those of CFD code. Bagheri and Ghasemi (2014) optimized a Wigley and a S60 vessel in the Froude numbers of 0.2 and 0.4 and reduced the vertical oscillations by 30%. Also Ghassemi et al. (2015) compared the results of the Savitsky method with a numerical method of ANSYS-CFX code for a planing craft with and without step and showed that both methods gave acceptable answers but the CFD code showed better results. Mansoori and Fernandes (2016) investigated the hydrodynamic effect of the interceptor to control the dynamic longitudinal instability. They used a combined CFD and semiempirical formulae of Savitsky. Ertogan et al. (2017) obtained a dynamic model of a high-speed craft with trim tabs/interceptors for surge and pitch motions and designed an automatic controller which adjusts the command signal to increase fuel efficiency, safety and comfort of passengers. They used a full-scale sea trial data along with a system identification method and artificial neural network. Sakaki et al. (2017) used genetic algorithm to optimize the drag force of a planing vessel using Savitsky's formulae. They used the geometry of planing vessel and trim tab data as input data. Also, the objective was to obtain the minimum resistance at each constant velocity and trim angle by changing the longitudinal center of gravity and deadrise angle.
Therefore in the present research, using the semi-empirical Savitsky and Lewandowski formulations and also with the Routh-Hurwitz stability criterion (Ogata, 2009), the longitudinal and transverse stabilities of a DTMB 62 model 4667-1 planing craft are investigated and also by using genetic algorithm method the resistance of the craft is optimized in each cruise speed in both constant projected area and constant beam based on satisfying the longitudinal, transverse and static stability criteria limits. Fig. 1 shows the schematic configuration and the forces which are acting on the body of a planing vessel.

Method of analyzing
In Fig. 1, f 0 , a, and c are the distances of thrust line to the center of gravity (COG), frictional drag line to chine, and normal hydrodynamic force position to the COG respectively. General force and moment balance equations can be written in the form of Eqs. (1)-(3). It should be noted that the effects of f 0 , a, and c are negligible (Faltinsen, 2005). (1) τ ε where is the angle between the keel and the horizontal axis (trim angle) and is the angle between the thrust line and the trim angle. Also, T, N, D f , and W are the thrust, hydrodynamic lift force, drag force on the hull along the trim angle, and the weight of the vessel, respectively. Furthermore, a planing vessel attitude can be investigated by semi-empirical methods such as the Savitsky method using the force balance equations and the Savitsky formulations (Savitsky, 1964;Faltinsen, 2005).

Fn
L ⩾ 0.9 The Savitsky formula can be used in the range (Savitsky, 1964). Also, ITTC method can be used to calculate the friction force (Faltinsen, 2005).
To investigate the longitudinal stability, the equations of motion in heave and pitch should be solved. Also for analyzing the transverse stability of a vessel the equations of yaw, roll and sway should be used. By setting exciting forces equal to zero, all motion equations are written in Eqs. (4)-(8): The parameters used in the above equations are defined in Table 1.  By using Laplace transform to longitudinal (Eqs. (4) and (5)) or transverse equations (Eqs. (6)-(8)) and letting the determinant of the coefficient matrix to be zero, the result leads to Eq. (9): It should be mentioned that for each set of equations, the coefficients (A jk , B jk , and C jk ) of Eqs. (4)-(9) have different quantities which can be found in references (Lewandowski, 1997(Lewandowski, , 2004Faltinsen, 2005;Castro-Feliciano et al., 2017).
Also with Routh-Hurwitz stability criterion (Ogata, 2009), the condition of Eq. (10) (Faltinsen, 2005) should be satisfied to have the Longitudinal Direction Stability (LDS) (Faltinsen, 2005) or the Transverse Direction Stability (TDS) (Lewandowski, 2004) To calculate the moments of inertia of the base geometry, AutoCAD software was used and the result is compared with the relations (11) to (13). The comparison showed a good agreement, therefore for the new geometries, relations (11) to (13) are used to calculate the moment of inertia in all directions (Deb, 1999).
Also to optimize the drag force with the genetic algorithm (Melanie, 1998;Clement and Bloun, 1963), the static and dynamic stability criteria limits should be satisfied. The change of the length, longitudinal position of the center of gravity and the deadrise angle of the vessel are the important parameters which are used in this research.

Longitudinal dynamic stability
To validate the semi-empirical relations, the drag force results of the Savistky formulation are compared with the experimental results of 4665, 46677 and 4669 hull vessels of Clement and Blount (1963) and also with prismatic geometry shape vessel of Fridsma (1969) in Fig. 2. It shows that the results, especially at higher speeds, are in a good agreement with the experimental data. It should be noted that the speed range of the Savitsky formula is 4.4 m/s and higher with respect to . or (Savitsky, 1964;Faltinsen, 2005). It can be concluded from Fig. 2 that in the present test cases the Savitsky method can predict the drag force in lower velocities with a good accuracy.
To validate the present code with the Savitsky relations in longitudinal dynamic stability, the results of starting porpoising phenomena velocities of the present work are compared with the experimental data of Fridsma (1969) in Table 2. At lower velocities, the oscillations are very small and 1 m/s difference between the present results and the experimental results are not significant, showing a fairly good agreement.
In ongoing sections, the effects of some important parameters on longitudinal dynamic stability of DTMB 62 model 4667-1 planing vessel whose specifications are presented in Table 3 are investigated. 3.1 Effect of aspect ratio (L/B) on the longitudinal dynamic stability To find the effect of aspect ratio on the longitudinal dynamic stability, the condition of stability (Eq. (10)) is calculated with four different values of L/B equal to 1.67, 2.5, 4.09 and 5 and the results are presented in Fig. 3. It should be noted that for each case the beam is constant and only the length of the vessel is changed, therefore the center of gravity position is also modified in each L/B.   3 shows that with the increasing aspect ratio the stable region of the vessel increases with a nonlinear behavior. Because of the increase in the volume of the vessel with the increasing aspect ratio in constant mass, the draft of the vessel and the trim angle decrease and the hydrodynamic force center moves to the direction of the center of gravity and the torque of the forces decreases. Therefore the stability region increases.

Effect of L cg /L on the longitudinal dynamic stability
To investigate the effect of L cg /L on the longitudinal stability region, Eq. (10) is calculated with four different values of L cg /L equaling to 0.221, 0.303, 0.385 and 0.448, respectively and the results are depicted in Fig. 4. Fig. 4 shows that, with the center of gravity moving to the front of the vessel, the trim angle decreases and the stability region increases, and thus the oscillation also decreases.

Effect of deadrise angle on the longitudinal dynamic stability
To investigate the effect of deadrise angle on the longitudinal stability region, Eq. (10) is calculated with four deadrise angles of 5°, 10°, 14.5° and 20° and the results are depicted in Fig. 5. It shows that with the increasing deadrise angle the stable region increases but to a relatively small extent. As the freeboard and beam of the vessel remain constant with the decreasing deadrise angle, the volume of constant mass vessel increases and the draft decreases, so the vessel reaches the planing phase in a lower velocity and may enter the unstable region sooner.
It should be mentioned that the obtained results of the longitudinal dynamic instability region in this paper are relatively close to the numerical results of Masumi and Nikseresht (2017).

Transverse dynamic stability
In this section the transverse dynamic stability is analyzed with the Lewandowski formulations (Lewandowski, 1997). For validating the present code the K cgmax /beam of present results are compared with the experimental data of Wellicome and Campbell (1984) and Lewandowski's results (Lewandowski, 1997) in Fig. 6. K cgmax is the maximum distance between keel to the COG in which the planing vessel remains stable. It shows that the results are in a good agreement with the experimental data and the maximum error is about 10% in the velocity of 4.5 m/s.

Effect of aspect ratio (L/B) on the transverse dynamic stability
The results of the stable region in transverse dynamic    Fig. 6. Comparison of the present code with the experiment (Wellicome and Campbell, 1984) and Lewandowski's results (Lewandowski, 1997). stability, using the Savitsky relations and the Lewandowski model are presented in Figs. 7 and 8 for three different values of L/B equal to 2.51, 4.09 and 5. It should be noted that for each case the beam is constant and only the length is changed, therefore the position of the center of gravity is also modified in each L/B. Also the draft of the vessel is depicted in Fig. 8. It shows that the draft of the vessel at first increases with the increasing velocity and after a peak it decreases again, also this peak occurs in higher speeds with the increasing aspect ratio. But the peak appears at a lower value with increasing the aspect ratio. One of the important parameters to reduce the unstable conditions in transverse stability is the wetted area and the draft of the vessel. As is depicted in Fig. 7, increasing the aspect ratio increases the stable region.

Effect of L cg /L on the transverse dynamic stability
The results of the stable region in transverse dynamic stability, using the Savitsky relations and the Lewandowski mode are presented in Fig. 9 with three different values of L cg /L equaling to 0.344, 0.448 and 0.533. Fig. 9 shows that increasing L cg /L can increase the possibility of going to the unstable region. Although the longitudinal position of the center of gravity does not have any serious effect on the roll motion, the yaw stability is very dependent on this parameter (Lewandowski, 1997). When the center of gravity moves to the front of the vessel the draft of the transom and the trim angle decrease, therefore the yaw and sway motion stability decrease.

Effect of deadrise angle on the transverse dynamic stability
The results of the stable region in transverse dynamic stability are presented in Fig. 10 with three different values of deadrise angles equaling to 10°, 14° and 20°. Fig. 10 shows that with the increasing deadrise angle the transverse stability region decreases. Decreasing the deadrise angle increases the draft and the wetted surface area, therefore the restoring force increases and it makes the vessel more stable.

Optimization
In this research for optimizing the drag force, a code on the basis of the genetic algorithm is generated. To validate the code at first an easy and common problem is solved. In this problem, the variables of Eq. (14) should be found such that the summation of the variables is equal to zero. It is obvious that all variables should converge to zero. At first, each variable is defined in a domain and starts the algorithm to optimize Eq. (14). Answers for X 1 to X 5 are found as 0.741216×10 -3 , 0.13208×10 -3 , 0.46405×10 -6 , 0.11352×    Yasin MASUMI, Amir H. NIKSERESHT China Ocean Eng., 2019, Vol. 33, No. 3, P. 365-372 10 -8 and 0.25723×10 -3 respectively and available results are shown. It should be noted that with the increasing iterations the answers will become closer to zero.
The ranges of design parameters are depicted in Table 4. The mass is selected 100 kg as in the original one. As the thrust angle in the Savitsky formulations is not very significant, the thrust angle in these calculations is also set to 10°a s in the original one.
The objective of optimization is to decrease the drag force in the following cruise speeds: 2. 12, 4.24, 6.6, 8.486 and 10 m/s. The optimizations are done in two cases, one at a constant beam (B=0.597) and the other at a constant projected area (A=LB=1.457 m 2 ). In the second case the beam cannot be lower than 0.3 m which is a necessary condition for seating in a vessel. Fig. 11 shows the flowchart of using the genetic algorithm to obtain the minimum drag force. Figs. A1 and A2 also show the present calculations flowcharts of the expanded optimization and Savitsky formula solver.

Length optimization
The results of length optimization for both of the abovementioned cases are shown in Fig. 12. It is interesting to see that in low and moderate speeds the optimum length is near the main length of 4667-1 vessel, but in higher speeds the optimum length decreases.  Fig. 14 gives the optimization results of deadrise angle in both cases of the constant beam and constant projected area. It shows that the deadrise angle can change from 4° to 9° and the average deadrize angle is near 7°, but in the original 4667-1 vessel the deadrise angle is near 14° to reduce the slamming force on the bottom of the vessel (Ghazizade-Ahsaee and Nikseresht, 2013).

Optimization of drag force in the cruise speeds
Now by using the above graphs the optimized drag force for each cruise speed can be achieved. It should be noted that the thrust line angle is chosen as 10° in these calculations. Figs. 15 and 16 show the drag force of optimized hull in each cruise condition (CC) versus velocity in the constant projected area and constant beam respectively. Figs. 12 and 13 show that the optimized L and L cg are in the same range in low velocities, therefore the optimized drag force graphs in Figs. 15 and 16 for the velocities under 6.6 m/s    are very close to each other. In high velocities, the optimized results are also close to each other, but in all optimized velocities the average drag force reduction is 22% and 18% in the constant projected area and beam respectively in comparison with the original hull form of 4667-1 vessel.

Conclusions
In the present study the effects of aspect ratio, the length of the center of gravity from the transom to the length of the vessel and the deadrise angle on the longitudinal and trans-verse stability of DTMB 62 model 4667-1 planing hull are investigated. Aiming at having a stable vessel, the drag force in each cruise velocity is optimized in constant projected area and beam respectively. The general conclusions of this research are as follows: (1) The used semi-empirical relations can predict the region of stability in a planing craft with a good accuracy.
(2) Decreasing the aspect ratio, the distance of the center of gravity from the transom to the length of the vessel and the deadrise angle decrease the stable region in longitudinal stability.
(3) Decreasing the aspect ratio, the distance of the center of gravity from the transom to the length of the vessel increases and the deadrise angle decreases the stable region in transverse stability.
(4) Using the genetic algorithm to optimize the drag force in constant speeds, one can reduce the average drag force in each optimized velocities by 22% and 18% in constant projected area and beam, respectively.
(5) Based on the optimization results at higher speeds, it is better to have a shorter hull to optimize the drag force in planing vessels.
(6) The deadrise angle of the original hull of the DTMB 62 model 4667-1 planing vessel is larger than the optimized value and it is dependent on the slamming force that acts on the bottom of the vessel.   Yasin MASUMI, Amir H. NIKSERESHT China Ocean Eng., 2019, Vol. 33, No. 3, P. 365-372