Ship hull optimization based on wave resistance using wavelet method

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2015,27(2):216-222 DOI: 10.1016/S1001-6058(15)60475-9

Ship hull optimization based on wave resistance using wavelet method* ZHAO Yong (赵勇)1,2, ZONG Zhi (宗智)3, ZOU Li (邹丽)3 1. Transportation Equipment and Ocean Engineering College, Dalian Maritime University, Dalian 116026, China 2. State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China, E-mail: [email protected] 3. State Key Laboratory of Structural Analysis for Industrial Equipment; School of Naval Architecture, Dalian University of Technology, Dalian 116024, China

(Received November 19, 2013, Revised January 21, 2014)

Abstract: The ship hull surface optimization based on the wave resistance is an important issue in the ship engineering industry. The wavelet method may provide a convenient tool for the surface hull optimization. As a preliminary study, we use the wavelet method to optimize the hull surface based on the Michel wave resistance for a Wigley model in this paper. Firstly, we express the model’s surface by the wavelet decomposition expressions and obtain a reconstructed surface and then validate its accuracy. Secondly, we rewrite the Michel wave resistance formula in the wavelet bases, resulting in a simple formula containing only the ship hull surface’s wavelet coefficients. Thirdly, we take these wavelet coefficients as optimization variables, and analyze the main wave resistance distribution in terms of scales and locations, to reduce the number of optimization variables. Finally, we obtain the optimal hull surface of the Wigley model through genetic algorithms, reducing the wave resistance almost by a half. It is shown that the wavelet method may provide a new approach for the hull optimization. Key words: hull optimization, wavelet method, Dauberchies wavelet, wave resistance, Michel theory, Wigley model

Introduction The wave resistance is very sensitive to the ship hull surface, which might be reduced by a large proportion through the hull optimization in a high-speed situation[1-3]. Therefore, the ship hull optimization is of a very important engineering significance. The ship hull surface must be expressed in a form, which is a prerequisite in the optimization. Normally, the hull surface or its derivative function is expressed by the non-uniform rational B-spline (NURBS) in many blocks due to complexity of the hull surface as in Refs.[4-

* Project supported by the Natural National Science Foundation of China (Grant Nos. 51309040, 51379033), the National Key Basic Research Development Program of China (973 Program, Grant No. 2013CB036101) and the Fundamental research fund for the Central Universities (Grant No. DMU3132015089). Biography: ZHAO Yong (1981-), Male, Ph. D., Lecturer Corresponding author: ZONG Zhi, E-mail: [email protected]

6]. In this paper, we express the whole hull surface by a wavelet expression and then derive a wave resistance formula in the wavelet space and optimize the hull surface aiming to reduce the wave resistance, as a new approach for the hull optimization. Since the 1980s, the study of the wavelet has made a great progress, especially by the work of Grossmann and Morlet, Dauberchies[7] and Taswell[8]. Since then, the wavelet method has found many applications in the field of signal and image processing. Due to wavelet bases’ unique properties, such as the compact support, the multi-scale resolution and the fast decomposition algorithm, the wavelet analysis enjoys great potential advantages in the numerical calculations as compared to the Fourier analysis. By expressing functions into the wavelet series, the information of the multi-resolution and the localization of the function can be easily identified, thus the wavelet can serve as a mathematical microscope. By setting a threshold for the wavelet coefficients, the local structure can be captured, while the global error is controlled by this threshold. The wavelet method is well suitable in solving those problems with its multi-scale property

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and local structure[9-11] . As a preliminary study, we firstly utilize the wavelet method to solve the wave resistance problem based on the Michel theory for a Wigley model and then to optimize the hull in this paper.

1. An outline of wavelet The wavelet method enjoys such properties as the compact support, the symmetry, the orthogonality, and the bi-orthogonality. Compared to the Fourier transform, the compact supported bases in the wavelet is a significant improvement. The wavelet transformation of a function converts the function into a series of small wave-like bases, allowing the function to be stored in an efficient way. The wavelet method has great advantages in solving the problems characterized by multi-scale and local structures, such as the turbulence and wave resistance problems. Here we will discuss some basic properties of the wavelet bases. For a given function f ( x) , we can measure the component with a scale a around the position b by the following process







x k ( x)dx = 0 , k = 0,1, 2,  , N  1

(3)

For example, if a polynomial is less than N order, its wavelet coefficients are zero if we choose a particular wavelet with an N order vanishing moment. This property leads to an effective compression in the wavelet method. In the practical numerical method, we often use discrete wavelet transformations. The bases of the wavelet are formed by dilation and translation of a mother wavelet function  ( x) as

 j , k ( x) = 2 j / 2 (2 j x  k )

(4)

where j , k  Z , j is the scale index, k is the location index. The function  ( x) has a companion, the scaling function  ( x) , and these functions satisfy the following relations: L 1

 ( x) = 2  g k  (2 x  k )

(5)

k =0

W f ( a, b) =  f ,  a , b ( x )  = 

+







f ( x)



1 a

f ( x) a , b ( x)dx =

 xb  dx  a 



L 1

 ( x) = 2  hk (2 x  k )

(6)

k =0

(1)

where  a , b ( x) is a wavelet function, with two basic

where g k = (1) k hL  k 1 , k = 0, , L  1 and  ( x) satisfies the unity condition 

parameters a and b , representing the scale and the position, respectively.  ( x) is the mother wavelet



function,  a , b ( x) is a translation with the shifting pa-

Similarly, the bases of the scale function are formed by dilation and translation of a mother scale function  ( x)

rameter b ( b  0 for rightward, b  0 for leftward) and the duration controlled by the parameter a (extended if a  1 , compressed if 0  a  1 ). After some particular post-processing, one can reconstruct function f ( x) in the following way 1 f ( x) = C



 



 0

2

a W f (a, b) a , b ( x)da db

(2)

where  | ˆ ( ) |2 d   , ˆ ( ) =   ( x)e  i x dx   | |

C = 



C   is called the admissibility condition. The vanishing moment is an important index. The wavelet has an N order vanishing moment under the condition that



 ( x)dx = 1

 j , k ( x) = 2 j / 2  (2 j x  k ) , k  Z

(7)

(8)

Two-dimensional bases can be formed by the tensor product of one dimensional bases in the following way,

 j , kl ( x, y ) =  j , k ( x) j , l ( y )

(9a)

 j , kl ( x, y ) =  j , k ( x) j , l ( y ) ,  = 1

(9b)

 j , kl ( x, y ) =  j , k ( x) j , l ( y ) ,  = 2

(9c)

 j , kl ( x, y ) =  j , k ( x) j , l ( y ) ,  = 3

(9d)

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The coefficients H = {hk , k = 0, L  1} and G = {g k , k = 0, L  1} are mirror filters. Once the filter H is chosen, the functions  and  are fully determined. For the Dauberchies wavelets[7], L  2M  1 . Let sup  and sup be the domains of the scaling function and the wavelet function, we have sup  = [0, 2 M  1] and sup = [ M +1, M ] . The Dauberchies wavelet does not have an explicit expression, but can be generated at any scale using the scale formula in an iterative manner. The wavelet bases provide us a multi-scale resolution to express a function. It is assumed that the function f ( x) can be well approached on the finest scaling index J , i.e., cJ , k = f ( xJ , k ) , and then it can be decomposed in terms of the sum of a series of wavelet base functions J 1

f J ( x) =  C j0 , k  j0 , k ( x) +   d j , k j , k ( x) k

j = j0

(10)

k

The coefficients are obtained by the wavelet decomposition formula: L 1

c j 1, k =  hl c j , l +2 k

(11)

l =0

f J ( x)  f dJ  ( x)  C1

(14)

And the number of the significant wavelet coefficients K is bound by  and the wavelet’s vanishing moment N as

  C2 1/ 2 N

(15)

The coefficients in Eqs.(14) and (15) depend on the wavelet vanishing moment and the function f ( x) . The threshold plays two roles: making the approximation adaptive, and controlling the approximation error globally. The similar situation can be simply extended to the multi-dimensional space by the tensor product. In the next section, we will have the ship hull surface’s wavelet expression and verify the accuracy by the Wigley model.

2. Hull surface’s wavelet expression Assume that the smallest scale information of the hull surface can be identified on scale J , denoted by f J ( x, y ) , with the initial scale coefficients cJ , kl = f ( xJ , k , y J , l ) , we might do the decomposition until the

L 1

d j 1, k =  gl c j , l +2 k

(12)

l =0

For functions that contain isolated small scales on a large scale background, most wavelet coefficients are small, thus we can retain a good approximation even after discarding a large number of wavelets with small coefficients. This situation can be explained by the wavelet’s vanishing moment. More precisely, if we rewrite the approximation as a sum of two terms composed, respectively, of wavelets with absolutely value beyond and below a prescribed particular threshold  , f J ( x) = f dJ  ( x) + f dJ  ( x)

(13)

largest scale is reached, which can be expressed by the following multi-scale wavelet expression 2 j0 1 2 j0 1

f J ( x, y ) =

 c

j0 , kl

 j , kl ( x, y ) + 0

k =0 l =0

J 1 2 j 1 2 j 1 3

d    

j , kl

j = j0 k =0 l =0

 j,kl ( x, y )

(16)

=1

The wavelet coefficients d j , k can be obtained by the two-dimensional decomposition formula. Now the ship surface is projected onto a sequence of nested spaces, so it can be analyzed in multi-scales. The scale coefficients cJ 0, kl at the coarsest scale J 0 represent the main information of the ship surface. The wavelet coefficients d j , k represent the detailed information of

where J 1

f dJ  ( x) =  c j0 , k  j0 , k ( x) + k

 

d j , k j , k ( x) ,

j = j0 d j , k 

J 1

f dJ  ( x) =

roximation error by the significant wavelets whose coefficient amplitude is above the threshold  is bounded by the following restriction:

 

d j , k j , k ( x)

j = j0 d j , k 

According to Vasilyev and Kevlahan[12], the app-

the corresponding scale. Similarly, we define an wavelet coefficient threshold such that the absolute value which are less than the threshold can be ignored, which brings about two benefits, a great reduction of the number of items in the ship hull surface expression, and the control of the reconstruction surface’s accuracy . For instance, the Wigley model can be expressed as

219

J 1 2 j 1 2 j 1 3

B  4 x 2  z2  y ( x, z ) = 1  2 1  2  2 L  H 

   d

(17)

f ,s j , kl

 sj , kl ( x, z )

(19)

j = j0 k =0 l =0 s =1

Here we choose B = 0.2 m , L = 2 m , H = 0.125 m. We approximate it in the finest scale J = 6 , repeating the decomposition at the coarsest scale J 0 = 2 with the Db2 wavelet. We choose the threshold  = 1.0  103 , the resulting 436 wavelet coefficients are kept instead of 4 096, which means that with less then 10% of the coefficients, the complete information can be captured in the finest scale J = 6 at an accuracy loss in the order of  = 1.0  103 , which can be verified by the error distribution as shown in Fig.1.

2 j0 1 2 j0 1

 c

g1 ( x, z ,  )  g1J ( x, z ,  ) =

g1 j0 , kl

 j , kl ( x, z ) + 0

k =0 l =0

J 1 2 j 1 2 j 1 3

   d

g1 , s j , kl

 sj , kl ( x, z )

(20)

j = j0 k =0 l =0 s =1

2 j0 1 2 j0 1

g 2 ( x, z ,  )  g 2 J ( x, z ,  ) =

 c

g2 j0, kl

 j , kl ( x, z ) + 0

k =0 l =0

J 1 2 j 1 2 j 1 3

   d

g2 , s j , kl

 sj , kl ( x, z )

(21)

j = j0 k =0 l =0 s =1

Using the orthogonal property of the wavelet bases of the same scale, we have





 j , k ( x) j , l ( x)dx =  kl

(22)

 j , k ( x) j , l ( x)dx =  kl

(23)









And the wavelet space orthogonal property says

Fig.1 Error distribution of reconstructed Wigley model

m 1

3. Hull optimization based on Michel theory in wavelet expression Based on the linear theory, the wave resistance can be calculated by the Michel formula in Gotman[13] F=

/2

0

( I2 + J2 )sec5 ( )d

(18)

Vm = V0  Wi

The integral calculation can be simplified as the product of the corresponding wavelet coefficients, 2 j0 1 2 j0 1 3

I =

  c k =0 l =0 s =1

where 0

L/2

h

L/ 2

I =  d z 

(24)

i =1

J 1 2 j 0 1 2 j 01 3 f j0 , kl

c

g1 j0 , kl

+

2 j0 1 2 j0 1 3

f ( x, z ) g1 ( x, z ,  )dx ,

J =



g1 ( x, z ,  ) = e  K0 z sec

2

g1 ( x, z ,  ) = e  K0 z sec

( )

cos[ K 0 x sec( )] ,

2

( )

cos[ K 0 x sec( )] ,

2

( )

sin[ K 0 x sec( )]

g 2 ( x, z ,  ) = e  K0 z sec

g1 ( x, z , ) , g 2 ( x, z , ) are related to the wave conditions and f ( x, z ) is the ship hull surface, which can be expressed in wavelet multi-scale forms as follows: 2 j0 1 2 j0 1

f ( x , z )  f J ( x, z ) =

 c k =0 l =0

f j0 , kl

 j , kl ( x, z ) + 0

f ,s j , kl

d jg,1kl, s (25)

J 1 2 j 0 1 2 j 01 3

  c jf0 , kl c gj02, kl + 

k =0 l =0 s =1

  d

j = j0 k =0 l =0 s =1

  d

f ,s j , kl

d jg,2kl, s (26)

j = j0 k =0 l =0 s =1

The ship hull surface’s wavelet coefficients are used as the optimization variables, according to the property that the wavelet function’s integral is equal to zero





 ( x)dx = 0



(27)

It is concluded that the volume does not change with the optimization variables. Denoting the change of the ship hull surface’s wavelet coefficients as d j ,fkl, s , we obtain the target function

220

F = 

/2

0

( I2 +  J2 + 2 I  I + 2 J  J )sec5 ( )d

(28) where J 1 2 j 0 1 2 j 01 3

 I =

   d

f , s j , kl

d jg,1kl, s ,

j = j0 k =0 l =0 s =1 J 1 2 j 0 1 2 j 01 3

 J =

   d

f , s j , kl

d jg,2kl, s

j = j0 k =0 l =0 s =1

In summary, we need to find the optimal solution f , s

d j , kl , to make the following expression minimum if f ( xi , z j )  0 . f ,s  {d j , kl } = d j , fkl, s 

Nk

 [( I )

2

+ ( I ) 2 + 2( I ) I +

k =1

2( J ) J ]sec5 ( k )  k  min}

(29)

Furthermore, we use the wavelet method’s features of the multi-scale resolution and the spatial localization to analyze each optimization variable’s contribution in the resistance, and then choose those with a significant contribution, thus reducing a large number of optimization variables.

Fig.2 Wave resistance coefficient for Wigley reconstruction at several scales

Fig.4 The distributions of the quadratic sums of Wigley ship’s wavelet coefficients at j = 5,4,3,2

Fig.3 Wave resistance coefficient by wavelet coefficients on separated scales

In what follows, we analyze the wave resistance distribution on scales and locations. For the scale index j , we reconstruct the ship surface on various scales, from the coarsest to the scale j ,

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Fig.5 Molded lines of Wigley ship after optimization 2 J 0 2 J0

f j ( x, y ) =  cJf0 , kl J 0 , kl ( x, y ) + k = l =1

J 1 2 j

2j

3

d   

 j , kl ( x, y ) , J 0  j  J

j , kl

j = J 0 k =1 l =1

(30)

=1

Substituting it into the wave resistance formula (21), we obtain the relation of the resistance coefficient versus the Froude number, as shown in Fig.2. To observe the resistance on a separated scale, we can consider the difference between the resistances of adjacent scales, for example, if we want to see the resistance due to the wavelet coefficients on the scale j = 2 , we might consider the difference between the resistances on the scales j = 3 and j = 2 . Figure 3 shows the wave resistance on each scale. It is shown that the wavelet coefficients on the scale j = 2 play an important role, thus we choose them as the optimization variables. Next we analyze the wave resistance’s spatial distribution by the wavelet coefficients’ spatial distribution. Figure 4 shows the sum of the squares of the wavelet coefficients in three directions, i.e., the horizontal, the vertical, and the diagonal directions, and it is found that the large values are localized near the ship head and the waterline, as is consistent with the real situation. Moreover, at the scale j = 2 , the amplitude is much larger than at other scales, and they are uniformly distributed. Based on the above observation, we select the wavelet coefficients on the scale j = 2 , totally 48 variables, to be the final optimization variables.

We use the genetic algorithm to optimize these 48 variables on the designed Fr = 0.3 , and the results are shown in Fig.5. The wave resistance is reduced by 49.6%. The resulting hull is flatter near the hull’s bow and stern, convex under the waterline, and then contracted sharply near the the waterline. The bottom is a clear plate and then a sharply inward contraction. The optimized hull transfers a partial volume from the water surface nearby to a certain distance below the waterline. However, this hull does not meet the smoothness requirement in real engineering sense. Also, the frictional resistance should be increased for this kind hull. It should be noticed that we obtain this optimal hull mainly based on the mathematical consideration by the wavelet method.

Fig.6 Comparison of wave resistance coefficients of Wigley hull and its optimal hull using wavelet approach by Michel theory calculating by Noblesse theory

In order to check the drag’s reduction, the wave resistance after optimization using the wavelet approach by the Michell theory is calculated by the Noblesse

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wave resistance theory for a slender ship[14], as shown in Fig.6. In the Noblesse theory, the original hull’s wave resistance coefficient is 1.8104, while the optimized hull’s coefficient is reduced to 1.2104, one third wave drag reduction at the designed Fr = 0.3 . The drag reduction proportion is less than the previous calculation. This result is reasonable due to the different features of the Michel and Noblesse theories. From these consistent results, we may see that the proposed model is rational and the calculation is accurate.

4. Conclusions Utilizing the excellent mathematical properties of the wavelet, such as the compact bases, the orthogonal subspace partition and the fast algorithm for decomposition and reconstruction[15,16], we find its application in the ship hull surface optimization by reducing the wave resistance in this paper. Firstly, the hull surface is expressed by the wavelets with a fine enough scale, secondly, the expression is approximated by introducing a threshold for the wavelet coefficients, thus a very concise approximate expression is obtained with an error of the order of the threshold, thirdly, the wave resistance formula is derived in the wavelet form, and fourthly, the wavelet coefficients of the hull surface are chosen as the optimization variables, while the hull volume keeps constant. A good understanding of the wave resistance’s distribution is obtained in terms of scales and locations in the wavelet language. The wavelet method provides a new approach for the hull optimization. As a preliminary study, we adopt the linear wave resistance theory and a simple hull model. The smoothness constraint is not included in the optimization process, so the optimal hull may not meet the actual requirements in ship engineering. We are considering a further research in terms of the non-linear wave resistance theory as in Refs.[17,18], and the practical hull surface and smoothness requirements.

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14] [15]

[16]

Acknowledgement This work was supported by the Open Research Fund of State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University (Grant No. 1402).

[17]

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