224

2011,23(2):224-233 DOI: 10.1016/S1001-6058(10)60107-2

NUMERICAL SIMULATION OF WAVE RESISTANCE OF TRIMARANS BY NONLINEAR WAVE MAKING THEORY WITH SINKING AND TRIM BEING TAKEN INTO ACCOUNT*

WANG Zhong, LU Xiao-pingCollege of Naval Architecture and Power, Naval University of Engineering, Wuhan 430033, China,E-mail: wangzhonghj@sohu.com

(Received October 8, 2010, Revised January 6, 2011)

Abstract: Up to now, there are no satisfactory numerical methods for simulating wave resistance of trimarans, mainly due to the difficulty related with the strong nonlinear features of the piece hull wave making and their interference. This article proposes a numerical method for quick and effective calculation of wave resistance of trimarans to be used in engineering applications. Based on Wyatt’s work, the nonlinear free surface boundary condition, the time domain concept, and the full nonlinear wave making theory,using the Rankine source Green function, the 3-D surface panel method is expanded to solve the trimaran wave making problems, with high order nonlinear factors being taken into account, such as the influence of the sinking and trim, transom, and ship waveimmersed hull surface. And the software is successfully developed to implement the method, which is validated. Several trimaranmodels, including a practical trimaran with a sonar dome and the transom, are used as numerical calculation samples, their wavemaking resistance is calculated both by the present method and some other methods such as linear (Dawson) methods. Moreover, sample model resistance tests were carried out to provide data for comparison, validation and analysis. Through the validation by model experiments, it is concluded that present method can well predict the wave making resistance, sinking and trim, and the accuracy of wave making resistance calculation is significantly improved by taking the trim and sinking into account, especially at high speeds.

Key words: trimaran, nonlinear wave making resistance, surface panel method, sinking, trim, ship model test

Introduction

The trimaran discussed in the present article consists of a slender center hull (also called the main hull) and two slender outer hulls of much smaller size (also called the side hulls or outriggers), as shown in Fig.1. In recent years, a considerable research effort has been put on the trimaran due to its many advantages that come from its special hull cons- truction, such as the excellent resistance performance, the high stability, the broad deck and satisfactory sea- keeping performance, and the trimaran is widely accepted as a new ship form with a wide application prospect[1,2]. A good prediction method for the wave making resistance of trimaran is very important for its

* Project supported by the National Defense Science Foundation Program (Grant No. 9140A14070306JB1114). Biography: WANG Zhong (1981-), Male, Ph. D.

practical development and building.

Fig.1 The coordinate on the trimaran

As was observed[3], the wave interference between the center hull and the outriggers is very complicated, with the waves repeatedly reflected and superimposed in a limited water area restrained by the main hull and the inner side wall of each outrigger. Moreover, the outrigger bow spray may be evident and mixed with the wave making flow at high speeds.

225

How and how much they are mixed varies with the outrigger position[4]. Therefore, the ship waves of trimarans are much more complex than ordinary monohulls, with much stronger nonlinearity. Conse- quently, theoretically, it would be much more difficult to calculate the trimaran wave resistance and its prediction remains an issue despite a number of articles on theoretical calculations and model tests of the trimaran wave resistance , which mainly published in the 1990s, in the USA, Britain, Japan, based on the ship form concept and the application viewpoints. Wilson et al.[5] presented design concepts of opti- mizing piecehull arrangement to minimize the wave resistance of multihull by the piecehull wave canceling consideration. Andrews and Zhang[6], based on studies initiated in UCL, explored the concept trimaran escort hull form performance, taking into account of the resistance, seakeeping performance, maneuverability, as in some extent, related to the demonstration trimaran RV Triton constrtuction. Suzuki and Ikehata[7] also carried out studies of the trimaran wave resistance performance based on the linear wave making theory and model tests, where the outrigger position for the minimum wave resistance was optimized by mathematical programming technique. In the same category as the above mentioned linear wave making theory, Han and Huang[1] put forward a new slender ship wave making theory to solve the trimaran wave making problems, as an earliest study in China about trimaran wave making field. Lu et al.[8], Li et al.[9] also developed the linear wave resistance theory for the trimaran wave resistance calculation and analysis. Although the wave resistance curve obtained from the linear wave making theory has a roughly correct trend, the calculation accuracy is not satisfactory due to the nonlinear effects, especially as compared with the conventional monohulls. In recent years, some semi-nonlinear wave making theory models were introduced to solve the trimaran wave making problems. Chen and Zhu[10], Lu et al.[11,12], employed the Rankine source method to calculate the wave resistance of a trimaran with transom stern by a semi-nonlinear method based on the double model velocity potential as implemented by Dawson. Wang and Zou[13] also employed the Rankine source method to trimaran wave resistance numerical calculation under the linear free surface condition of uniform flow velocity. In their earlier studies, Kang et al.[14], Kim et al.[15], applied the double model semi-nonlinear Rankine source method as Dawson’s to the trimaran wave resistance calculation and analysis. The calculation accuracy, especially, the wave resistance curve trend obtained by using the semi-nolinear methods for trimarans were improved, nevertheless, the wave resistance still somehow deviates from the model test results, more than for conventional monohulls, especially, for the

trimaran with the transom. Therefore, the present article will focus on the full nonlinear wave resistance numerical calculation method for the trimarans.

In the nonlinear wave resistance calculation, Rankine source methods were used for conventional monohulls[16-20]. In the present article, Wyatt’s nonlinear wave making calculation form will be employed as the foundation of the mathematic model and calculation method[16], the NURBS is introduced to express the ship hull form, while the instantaneous hull wetted surface is determined along with the wave profile, sinking and trim calculation results, and the quadrilateral meshes are generated on the instan- taneous wetted surface by the paving method[4], where the instantaneous hull surface boundary condition is satisfied accurately in each iteration calculation step, both Raven’s optimized four-point upward difference scheme and the collocation-point shift method are adopted to satisfy the radiation condition, while the desingularization is realized by raising the panel a distance above the collocation-point[21], which would also enhance the stability of the iteration.

The calculation by the present procedure is carried out for the sample models of the Wigley piece hull trimarans and a transom trimaran of the sonar dome, and the comparison between the calculation results and the model test results shows that the present methods can well predict the wave resistance, sinking and trim for trimarans, moreover, the accuracy of the wave resistance calculation can be significantly improved by taking the trim and sinking into accounts, especially at high speeds.

1. Theoretical basis 1.1 Mathematical models and numerical method

Consider a ship with a steady forward speed, U ,in the presence of the free surface. The coordinate system is fixed on the ship hull, as shown in Fig.1, with the forward direction as positive x direction, and upward direction as positive z direction, and with the origin at the middle point of the center hull water plane symmetry axis line, so that, the positions where = 0z correspond to the calm water free surface. With the incompressible, inviscous, irro- tational flow in mind, we define the velocity potential Φ , which satisfies Laplace’s equation throughout the fluid domain

2 = 0Φ∇ , for all ( , )z x yζ≤ (1)

where ζ is the free-surface elevation, i.e., the ship wave surface elevation.The kinematical free-surface boundary condition can be written as

+ + =t x x y y zζ Φ ζ Φ ζ Φ on = ( , )z x yζ (2)

226

and the dynamical free-surface boundary condition can be written as

21 1+ + =2 2t g UΦ Φ Φ ζ∇ ∇ on = ( , )z x yζ (3)

The velocity potential Φ is divided into the free-stream potential, Ux− , and the wave making perturbation potential, φ , then Φ is expressed as

= +UxΦ φ− (4)The boundary condition on the hull surface can

be written as

= xUnφ∇ n (5)

where n is the unit normal vector on the hull surface, with the direction into the fluid domain, xn the component of n in x direction.

By substituting Eq.(4) into Eqs.(2) and (3), collecting linear terms on the left side, and employing the numerical discrete procedure of forward difference for the two equations[16], the kinetic free-surface boundary Eq.(2) and the dynamic free-surface boundary condition Eq.(3) become

+1 +1 +11 1=n n n nx zU

t tζ ζ φ ζ− − −

Δ Δ

( ) ( )1 1 1 11 +2

n n n n n n n nx x y y x x y yφ ζ φ ζ φ ζ φ ζ− − − −− − (6)

+1 +1 +11 1+ =n n n nxU g

t tφ φ ζ φ− −

Δ Δ

1 11 1 1+2 2 2

n n n nφ φ φ φ− −∇ ∇ ∇ ∇ (7)

Take the limit of tΔ → ∞ in Eqs.(6) and (7), then the time-dependent terms vanish, the discrete kinetic free-surface boundary Eq.(6) and the discrete dynamic free-surface boundary condition Eq.(7) are transformed to

( )+1 +1 1= + +2

n n n n n nx z x x y yUζ φ φ ζ φ ζ− − −

( )1 1 1 1+n n n nx x y yφ ζ φ ζ− − − − (8)

+1 +1 1 1+ = +2 2

n n n nxU gφ ζ φ φ− − ∇ ∇

1 112

n nφ φ− −∇ ∇ (9)

The superscripts 1n − , n , +1n in Eqs.(6)-(9) indicate the time iteration steps. 1.2 Sinking and trim determination

The ship sinking and trim is a two-dimensional motion of a rigid body, therefore, can be described by two variables, the vertical position of the dynamic buoyancy center and the ship trim angle[22]. The coefficients s and q in the formula d = +z s qxrefer to such variables, and they are solved from a pair of equations for the ship hull force equilibrium, i.e. , the equilibrium between the hydrodynamic pressure variation and buoyancy variation, and the equilibrium between the moments of these force variation, as expressed by the equations

0 0

( ,0)d + ( ,0)d = dL Lx x

p zx xS

s y x x q xy x x C n SΔ (10)

0 0

2( ,0)d + ( ,0)d = dL Lx x

p zx xS

s xy x x q x y x x x C n SΔ (11)

where the subscription L refers to the length of the ship water lines, S the ship hull wetted surface,

pCΔ the pressure variation, and zn the component of n in z direction. Then the values s and qcan be determined by iterations. 1.3 Transom stern flow simulation

To simulate the flow behind the transom stern, we set the kinetic boundary condition at the transom corner (bottom edge). Using a Taylor series expansion, the wave elevation derivative xζ can be expressed in the stencil of second-order accuracy at the panel j ,immediately aft of the transom (TR) based on the knowledge that the wave elevation on the immediate upstream panel +1j , is just TRξ , which expresses the vertical ordinate of the transom edge. So that the jth wave elevation x-derivative xξ is expressed as[16]

1 1 TR TR= + +x j j j jc c cζ ζ ζ ζ− − (12)

where

( )2

2

1=+j

fcf f x

−−Δ

, ( )2

1 2=

+jfc

f f x− −Δ

,

( )TR 2

1=+

cf f xΔ

,

227

xΔ is the longitudinal distance between surface collocation points, and f xΔ is the distance (segment length) forward from the jth panel center, corres- ponding to the stern edge elevation (wave elevation

TRζ ). According to Wyatt’s work, the best calculation results can be obtained when = 0.55f .1.4 Numerical method

The steady solution based on the mathematical models outlined above can be obtained by distributing the Rankine panel source on both the wetted hull surface and the free surface, as slightly different from Wyatt’s method[16]. To satisfiy accurately the non- linear boundary condition in the present wave making mathematical models, an iterative numerical calcul- ation procedure is necessary, where the Raven’s optimized four point upward difference scheme is employed[22] for the free surface condition equation as a fundamental numeration skill, and the free-surface panel collocation points are shifted 0.25 xΔ forward to satisfy the radiation condition, while the free surface panels are raised 0.8 xΔ above the collo- cation point for desingularization and the iteration process robustness.

Fig.2 Program flowchart

Initially, the nonlinear terms are set to be zero, and the solution for the linear equations is used as the full nonlinear problem’s initial solution, corres-

ponding to the semi-nonlinear solution similar to Dawson methods. The nonlinear terms are determined and introduced in the following iteration steps, where the linearization equation in matrix form is solved with the updated right-hand side vector. The process is repeated until the steady nonlinear solution is obtained, subsequently, the sinking and trim coefficients are determined by Eq.(10) and Eq.(11). Moreover, the updated influence matrix (the left side square array) is recalculated taking into account the sinking and trim and the wave profile to determine the wet hull surface and deformed free surface to complete an entire iteration step, and the process is repeated with the nonzero nonlinear terms obtained from the previous iteration steps. The program flowchart is shown in Fig.2, where the convergence of the inner iteration is measured by the error in the kinetic and dynamic boundary conditions, the convergence of the outer circulation in the free surface relaxation iteration is measured by the source strength distribution, sinking and trim, and wave elevation variation between the two iteration steps.

2. Numerical calculation results and discussions 2.1 Wigley hull trimarans

The Wigley piece hull trimarans are selected as the first group of calculation samples[23,24], both the main hull and outriggers of which are represented by Eq.(13)

2

2

= 1 12

2

B x zyL T

− − (13)

where L is the water line length, B the breadth, and T the draft. The main ship form parameters are listed in Table 1.

Table 1 Main ship form parameters Main hull Side hull Total

Length L 5 m 1.842 m 5 m

Breadth B 0.4 m 0.147 m 1.647 m

Draft T 0.178 m 0.066 m 0.178 m

Displacement 158.22 kg 7.94 kg 174.1 kg

Wetted surface 2.3629 m2 0.3207 m2 3.004 m2

/L B 12.5 12.5

/B T 2.25 2.25

bC 0.44 0.44

Two trimaran configurations[23,24] from the above Wigley trimarans are chosen as samples, Trimran-1,

228

with = 0.75 mp , = 0 ma , and Trimran-2, with = 0.75 mp , = 1.5 ma , where p refers to the

transverse clearance distance between the main hull centerline planes and the outrigger centerline, and athe longitudinal stagger distance between the center of the main hull and the center of the outrigger.

The free-surface mesh domain varied with calculation speeds. As a numerical demonstration, the mesh domain is extended 1.5 ship length downstream, 0.5 ship length upstream, and 1.0 ship length in the y-direction, at the speed of Froude number = 0.41Fr .The free-surface mesh is generated by a pre-developed program[25] with about 25 segments per wave length in the longitudinal direction, when 0.2Fr ≥ , the same mesh as when = 0.2Fr is employed to avoid too many panels beyond the computational resource, when 0.2Fr < . In the y-direction, the length of the transverse mesh intervals is increased at the free surface field points far away the center line(x axis) with a ratio of 1.05, outside the outrigger, and the minimum 1yΔ is about 1.9 percent of the center ship hull length. The free-surface meshes in the numerical calculation, as = 0.4Fr , are shown in Fig.3.

Fig.3 Free-surface mesh of Trimaran-2 for calculation, at = 0.4Fr

The number of panels on the trimaran hull wetted surface, including the wave immersion hull, is about 2 000 and they are continually reconstructed during the iterations, for the hull wetted surface would conti- nually vary with the wave profile variation. One of the ship wetted surface mesh configuration during the iteration procedure, as an illustration sample, is shown in Fig.4.

Fig.4 A sample of the hull wetted surface and the mesh configuration of Trimaran-2 during the iteration procedure

The resistance experiments for the Wigley

trimaran model were carried out in a towing tank of 510 m in length, 6.5 m in width and 5.0 m in depth, in China Special Vehicle Research Institute in June, 2006[23] , and again in September, 2009, with the Froude number in the range of 0.2 - 0.829, with respect to the main hull length, = /Fr U gL . All measurements, regarding sinking and trim, were conducted with the model in a free condition.

The comparison of the different numerical calculation results and the model test results is shown in Figs.5 and 6 (respectively, for Trimaran-1 and Trimaran-2 models), and the main insight may be summed up as follows.

Fig.5 Comparison between the calculated results and the model test results of the Trimaran-1

229

Fig.6 Comparison between the calculation results and the model test results of the Trimaran-2

(1) The theoretically predicted wC results from Dawson method under the linear free-surface condi- tion are similar to those of Ref.[22], referred as the linear method in Figs.5 and 6 or in the context, and the results of the present nonlinear method with consi- deration of the sinking and trim are nearly the same as those without consideration of the sinking and trim, when 0.3Fr < , however, the difference between the

wC calculation results including the posture (sinking and trim) and those not including the posture is evident, when 0.3Fr > , especially at high speeds. From Fig.5 only for Trimaran-1, the calculation results taking into account of the sinking and trim are much more consistent with the model test results, which strongly indicates the necessity of including the

sinking and trim for predicting the high speed tri- maran wave resistance.

(2) Figs.5 and 6 show that both the linear method and the present nonlinear method can predict the sinking and trim very well, when 0.4Fr < , while the present nonlinear method makes much better predi- ction for sinking and trim than the linear method when

0.4Fr > . The results show the advantage and make a validation of the present nonlinear method in compa- rison to the linear method, for the sinking and trim coefficients are directly related to the pressure distri- bution on the trimaran hull surface, furthermore, directly to the wave resistance of the trimaran.

Fig.7 Spray of the trimarans at the highest model test speed

(3) As far Fig.6 goes, the calculated wC results with the present nonlinear method are much lower than the rC model test results of the Trimaran-2 at high speeds, and it seems, even more unsatisfactory than those of the linear method. To examine the results of Fig.6 more carefully, we present the photographs of the model tests at a high speed as shown in Fig.7(a) for Trimaran-1 and in Fig.7(b) for Trimaran-2. A very strong water spray may be observed in Trimaran-2 situation, much stronger than in Trimaran-1 situation, and all wave resistance theory mentioned above have not included the related spray resistance, which explains the deviation of the present nonlinear calculation curves from the model test curves in the bulk portion and with the form resistance as in the minor portion of difference. On the other hand, comparing the calculation wave patterns in Fig.8 and Fig.9, the fore hull of Trimaran-2 outriggers

230

is seen to be located more at the wave crest area of the center hull as = 0.714Fr , in Fig.9(b), which can explain very strong water spray as shown in Fig.7(b). Now, from Figs.5-9, we see that the (a) stronger the outrigger water spray, the bigger the deviation of the present nonlinear calculation curves from the model test curves, and (b) when the spray resistance and the form resistance are reduced from the residual resistance coefficient rC , the calculated wC with the present nonlinear method would agree with the model tests the best, (c) so that the present nonlinear calculation method with consideration of the sinking and trim is in fact superior to the linear methods.

Fig.8 Wave pattern of trimarans when = 0.429Fr

Fig.9 Wave pattern of trimarans when = 0.714Fr

(4) Beside the validation of the accuracy of the wave resistance theory mentioned in (3), the other main purpose of providing wave patterns in Fig.8 and Fig.9 is to validate the present full nonlinear wave making theory for trimarans directly from the wave pattern theory calculation results. The theoretical wave patterns as shown in Fig.8 and Fig.9 are very similar to the model experiment wave patterns, as

shown repeatedly by the comparison with experiment survey wave patterns. Moreover it can be seen that in the trimaran configuration with the outrigger bow located near the center hull wave trough, as shown in Fig.8(b) for Trimaran-2, one may expect smaller wave making resistance; in contrast, in the trimaran configuration with the outrigger bow located near the center hull wave crest, as shown in Fig.9(b) for Trimaran-2, the resistance would be increased. These are validated by comparing the wave patterns incur- porating wave resistance curves and are also consis- tent with the results in Refs.[24,26,27] by the present authors. So, in fact, here is an effective measure to design the outrigger position arrangements. 2.2 Trimaran with bulbous bow and transom

A concept trimaran, called Trimaran-3, is taken as the third calculation sample. The main ship hull parameters are listed in Table 2, the ship hull line plan is shown in Fig.10, and the outrigger position para- meters are / = 0.3148a L , / = 0.0663p L , where a is the outrigger longitudinal stagger, p the outrigger transverse clearness, the same as in Fig.1. The Trimaran-3 outrigger stern edges are aligned to the center hull stern edge, according to the present values of outrigger parameters a , .p

Table 2 Main ship form parameters Main hull Side hull Total

Length L 135 m 50 m

Breadth B 10.4 m 1.8 m

Draft T 5.05 m 2.62 m

Displace- ment 3 451.45 t 112.10 t 3 675.65 t

Wetted surface 1 782.35 m2 227.73 m2 2 237.81 m2

/L B 12.98 27.78

/B T 2.06 0.69

bC 0.487 0.475

Fig.10 Trimaran-3 ship hull plan

One main purpose of taking Trimaran-3 as the calculation sample is to validate that taking into account of the transom influence, sinking, trim, and

231

others in the present nonlinear method, is important for the wave resistance calculation for high speed trimarans with the transom. The full nonlinear calculation results of wC are compared with the model test results of rC and Michell thin ship linear theory calculation results, as shown in Fig.11, where the model tests for rC are the towing tests under the free condition including sinking and trim. Fig.11 indicates that the Trimaran-3 wave resistance is well predicted with the present full nonlinear method. (1) The trend of the wC curve obtained by the nonlinear theory coincides with the test rC curve very satis- factorily, particularly, as compared with that obtained by the linear theory in the crest and trough points, (2) the deviation of the wC curve obtained by the nonlinear theory from the test rC curve is mainly due to the spray resistance (bulk portion) and form resistance (minor portion), so that the present full nonlinear theory method may be further developed to be a very accurate and practical technique for wCprediction for trimarans, by suitably handling the spay resistance and the form resistance.

Fig.11 Comparison between the calculated results and model test data of Trimaran-3

Another main purpose of taking up Trimaran-3 is to examine the rationality of the calculated wave patterns, as shown in Fig.12. Generally speaking, the wave patterns shown in Fig.12 are very reasonable, as compared with the transom trimaran survey wave patterns and the transom single hull survey wave patterns, for example, the transom single hull DTMB5415 wave patterns. As a feature of the transom single hull wave pattern, the stern wave trough is not obvious[18], as clearly shown in Fig.13 for the standard transom single hull DTMB5415. That feature comes from transom trimarans and is exactly reflected in Fig.12. Now that the theoretical wave patterns are validated, they can be applied to design the favorable outrigger position arrangements. From

Fig.12, it is shown that when = 0.387Fr and = 0.424Fr as in (a) and (b), the Trimaran-3

outrigger position arrangement is very favorable and the wave resistance is smaller than others, however, when = 0.566Fr as in (c), the outrigger position arrangement will be in an unfavorable condition. These discussions about the favorable outrigger position arrangement selection for transom trimarans according to the nonlinear theory wave patterns are also consistent with the results in Ref.[24,26,27] by the present authors.

Fig.12 Theoretical wave pattern of Trimaran-3

Fig.13 Theoretical wave pattern of DTMB5415 at = 0.41Fr

232

3. Conclusions The present article presents a three-dimensional

Rankine source surface panel method, including software for the trimaran wave problem solution, based on the full nonlinear wave making theory similar to Wyatt’s, taking into account of such high order nonlinear factors as the sinking, trim, transom flow, ship wave immersion hull, and others. Two Wigley trimarans, a trimatan with the transom and sonar dome, are selected as calculation samples. The wave making problems are solved by the present full nonlinear theory and validated by the model test results. The main conclusions are as follows:

(1) The present full nonlinear wave making theory for the trimarans can well predict the wave resistance, trim, sinking, both for the Wigley piece hull trimarans and for the practical transom trimarans with the sonar dome, the accuracy of which is better than the linear methods.

(2) The precision of the trimaran wave resistance calculation is further improved by considering the trim and sinking, especially at high speeds.

(3) It is necessary to investigate the outrigger spay resistance and the form resistance of the trimarans, which contribute much to the trimaran residual resistance and must be taken into account in the validation of the trimaran wave resistance calcu- lation results by the model test residual results.

(4) One may design the favorable outrigger position arrangement for trimarans according to the present nonlinear theory wave pattern.

AcknowledgementsThe authors wish to express their most sincere

appreciation for the valuable support from China Special Vehicle Research Institute Towing Tank, in terms of the experiment preparation, facilities pro- viding and the assistant jobs from its staff members.

References

[1] HAN Kai-jia, HUANG De-bo. The computation method of the wave resistance on the tramarans[J]. Journal of Harbin Engineering University, 2000, 21(2): 6-10(in Chinese).

[2] MIZINE I., AMROMIN E. and CROOK L. et al. High-speed trimaran drag: Numerical analysis and model tests[J]. Journal of Ship Research, 2004, 48(3): 248-259.

[3] LI Pei-yong, QIU Yong-ming and GU Min-tong et al. Experimental investigation on resistance of trimaran[J]. Shipbuilding of China, 2002, 43(4): 6-12(in Chinese).

[4] WANG Zhong. Nonlinear wave resistance numerical research and resistance characteristics analysis and resistance reduction for trimarans[D]. Ph. D. Thesis,

Wuhan: Naval University of Engineering, 2010(in Chinese).

[5] WILSON M. B., HSU C. C. and JENKINS D. S. Experiments and predictions of the resistance characteristics of a wave cancellation multihull ship concept[C]. 23rd American Towing Tank Conference.USA, 1993, 103-112.

[6] ANDREWS D. J., ZHANG J. W. Consideration in the design of a trimaran frigate[C]. Int. Symposium on High Speed Vessles for Transport and Defence. London, UK, 1995.

[7] SUZUKI K., IKEHATA M. Studies on minimization of wave making resistance for high speed trimaran with small qurtriggers[J]. Journal of Naval Architect Sociaty of Japan, 1995, 177: 113-122.

[8] LU Xiao-ping, LI Yun and DONG Zu-shun. Comparison of resistance for several displacement high performance vehicles[J]. Journal of Hydrodynamics, Ser. B, 2005, 17(3): 379-389.

[9] LI Pei-yong, QIU Yong-ming and GU Min-tong et al. Resistance analysis and calculation of trimaran with super-slender hulls[J]. Ship Engineering, 2002, (2): 10-12(in Chinese).

[10] CHEN Jing-pu, ZHU De-xiang and HE Shu-long. Research on numerical prediction method for wavemaking resistance of catamaran/trimaran[J]. Journal of Ship Mechanics, 2006 10(2): 23-29(in Chinese).

[11] LU Xiao-ping, WANG Zhong and SUN Yong-hua et al. Solution to wave resistance to trimaran using Rankine source method of Dawson type[J]. Journal of Huazhong University of Science and Technology (Natural Science Edition), 2008, 36(11): 103-107(in Chinese).

[12] WANG Zhong, LU Xiao-ping and MA Xiao-gang. Solution to the trimaran wave making problem with free-surface calculation domain transformation[J]. Chinese Journal of Hydrodynamics, 2009, 24(4): 486-492(in Chinese).

[13] WANG Hu, ZOU Zao-jian. Numerical research on wave-making resistance of trimaran[J]. Journal of Shanghai Jiaotong University (Science), 2008, 13(3): 348-351.

[14] KANG Kuk-Jin, LEE Chun-Ju and KIM Do-Hyun. Hull form development and powering performance characteristics for A 2500 ton class trimaran[C]. Practical Design of Ships and Other Floating Structures. Shanghai, China, 2001, 151-157.

[15] KIM D. H., VAN S. H. and KIM W. J. et al. Estimation of the optimum position for the side hulls of a trimaran by panel method[C]. 4th J-K Joint Workshop on Ship and Marine Hydrodynamics. Fukuoka, Japan, 1999, 67-71.

[16] WYATT D. C. Development and assessment of a nonlinear wave prediction methodology for surface vessels[J]. Journal of Ship Research, 2000, 44(2): 96-107.

[17] WANG Zhong, LU Xiao-ping and WANG Wei. A solution to wave making problem using fully nonlinear free-surface boundary condition[J]. Journal of Huazhong University of Science and Technology (Natural Science Edition), 2009, 37(8): 112-115(in Chinese).

[18] WANG Zhong, LU Xiao-ping and WANG Wei. Nonlinear wave making resistance calculation for transom-stern ship with consideration of sinkage and

233

trim[J]. Chinese Journal of Hydrodynamics, 2010,25(3): 422-428(in Chinese).

[19] ZHANG Bao-ji, MA Kun and JI Zhuo-shang. The optimization of the hull form with the minimum wave making resistance based on Rankine source method[J]. Journal of Hydrodynamics, 2009, 21(2): 277-284.

[20] CHEN Jing-pu, ZHU De-xiang. Numerical simulations of wave-induced ship motion in time domain by a Rankine panel method[J]. Journal of Hydrodynamics,2010, 22(3): 373-380.

[21] RAVEN H. C. Invicid calculations of ship wave making-capabilities, limitations and prospects[C]. 22ndSymposium on Naval Hydrodynamics. Washington DC, USA, 1998, 892-906.

[22] BRIZZOLARA S., BRUZZONE D. and TINCANI E. Automatic optimization of a trimaran hull form configuration[C]. 8th International Conference on Fast Sea Transportation(FAST 05). Saint Petersburg, Russia, 2005, 136-145.

[23] LI Yun, LU Xiao-ping. An investigation on the resis-

tance of high speed trimarans[J]. Journal of Ship Mechanics, 2007, 11(2): 191-198(in Chinese).

[24] WANG Zhong, LU Xiao-ping and FU Pan. Research on the effect of the side hull position to the wave making resistance characteristics of the trimaran[J]. Ship- Building of China, 2009, 50(2): 31-39(in Chinese).

[25] WANG Zhong, LU Xiao-ping and WANG Wei. Research on the free-surface mesh fast generation for the calculation of the trimaran wave making resistance[J]. Journal of Harbin Engineering University, 2010, 31(4): 409-413(in Chinese).

[26] WANG Zhong, LU Xiao-ping and WANG Wei. Application of the nonlinear wave making numerical method in the high-speed trimaran side hull position optimization[J]. Journal of Ship Mechanics, 2010,14(8): 863-871(in Chinese).

[27] WANG Zhong, LU Xiao-ping and WANG Wei. Research on regression and prediction of the trimaran resistance[J]. Journal of Ship Mechanics, 2010, 14(4): 355-361(in Chinese).