Fast Evaluation of Time-Domain Green Function for Finite Water Depth

China Ocean Engnering , Vol.17 , No.3 , pp.417- 426.C 2003 China Ocean Press , ISSN 0890-5487Fast Evaluation of Time-Domain Green Functionfor Finite Water Depth'TENG Bin(滕斌) ,HAN Ling(韩凌)and GOU Ying(勾莹)The State Key Laboratory of Coastal and Offshore Engineering ,Dalian University of Technology , Dalian 116024 ,China( Received 30 March 2003 ; accepted 22 May 2003 )ABSTRACTFor computation of large amplitude motions of ships fastened to a dock , a fast evaluation scheme is implemented forcomputation of the time-domain Green function for finite water depth. Based on accurate evaluation of the Green functiondirectly , a fast approximation method for the Green function is developed by use of Chebyshev polynomials. Examinationsare carried out of the accuracy of the Green function and its derivatives from the scheme. It is shown that when an appro-priate number of polynomial terms are used , very accurate approximation can be obtained.Key words : Green fiunction ; time domain ; fite water depth1. IntroductionShips restrained by cables and fenders in front of docks undergo large amplitude non-harmonicmotions in waves. For this kind of non-harmonic problem , a time-domain method must be applied. Linand Yue( 1990 ) used an integral equation with the time -domain Green function for infinite water depthto compute the ship motion in deep water. But for the present problem , the integral equation with thetime-domain Green function for finite water depth must be applied.The Green function is a field with a certain boundary and initial conditions produced by a sourceat a given point. It is the fundamental element of most analytical and numerical techniques for predict-ing wave motion induced hydrodynamic forces and pressures on floating or submerged bodies.The solution by the Green function to the hydrodynamic problem was first presented in 1960' s.However , the Green functions are typically expressed by infinite integrations in water wave problems.The integrands corresponding to the Green function are oscillatory and decay slowly with the increase ofintegrating variables. Thus , the complexity of the integration is the principal difficulty and not amena-ble to computations directly . Although the present computers run at a very high speed , appropriate al-gorithms are still very important for computing the. Green functions , which are used time after time in中国煤化工boundary element methods .It is very important to obtain the Green functMHCNMH Gyand rapidly for solvinghydrodynamic problems of ships. In the usual formulation , the time-domain Green function can be ex-* The present work was financially supported by the National Natural Science Foundation of China ( Grant No.50025924 )Corresponaing数盾. E-mail : bteng@ dlut. edu. cnTENG Bin et al ./China Ocean Enginering ,1X3), 2003 ,417- 426pressed as a multiply integration , and an integral equation is established to solve the velocity potentialfunction of water wave problems at each time step. Thus the burden of computation is very large and itis necessary to seek a fast evaluation method to compute the Green function. Based on the work of We-hausen( 1967 ), Finklestein( 1957 ), and Stokes( 1957 ), many researchers have developed variousalgorithms to deal with the Green functions in order to save computation time and memories of comput-ers. One of those methods proposed is to expand the Green functions in effective series . Newman( 1987a ) used this kind of method for the evaluation of a single integration in the potential of a sub-merged source that moves at a constant speed. Another method is to approximate the Green functions bytruncated Chebyshev expansions or the equivalent economized polynomials. Newman( 1985a ) outlinedthis kind of numerical methods for the computation of the velocity potential and its derivatives of linear-ized three- dimensional wave motions due to a unit source with harmonic time dependence beneath thefree surface where the fluid is of either infinite or constant finite depth. Newman( 1987b ) also used .this method to study numerical approximations of a double integration in terms of three-dimensionalpolynomials , which greatly facilitates the computation of the double integration. A suitable combinationof the ascending power series and asymptotic expansion provides better algorithms for evaluating theGreen function. Noblesse( 1982 ) used this kind of method to study the Green function concerned withthe theory of potential flow about a body in regular deep water waves , and obtained an asymptotic ex-pansion and a convergent ascending- series expansion for the Green function from two alterative com-plementary near-field' and far-field'. An analogous method was used by Huang( 1992 ) and Duanand Dai( 1996 ) for calculating the three-dimensional and two-dimensional time- domain Green functionsfor infinite water depth respectively . These techniques are substantially faster than conventional directmethods based on numerical integration .Presently , the research on the three- dimensional time-domain Green function for finite waterdepth is less to be seen. For computing ship motions in front of docks , the present work is to providea new computational approach based on the use of multidimensional polynomial approximations , andthis approach greatly reduces the computing cost in numerical evaluation of the Green function. To ac-celerate the computation further , Chebyshev polynomials can be converted into simpler equivalent ordi-nary polynomials. In the whole domain the Green function is rapidly oscillatory and includes singulari-ty. It is not very effective to approximate the Green function in the whole domain directly. The ap-proach used in the paper is to divide the physical domain into several zones , and use different approxi-mations in each zone. The polynomial approximation is compared with the direct computation ap-proach. It is found that a desired accuracy can be. achieved bv the algorithm of Chebyshev polynomial中国煤化工approximation with not too many terms .HCNM HG2. Analytic Formulas for the 1ime-Domain Green FunctionFor diferent problems , the Green function of different foms is adopted. The present work is con-cemed with the three-dimensional time-domain Green function for simulating the real time non-harmon-ic motions万有整据g bodies in finite water depth. In a fluid of depth h , the velocity potential at timeTENG Bin et al ./China Ocean Enginering ,1X3), 2003 ,417-426419t due to an impulsive source at time τ is represented in the following form by Wehausen and Laitone( 1960 )Q( P,t ;Q ,t)=8t). 8(t)_28(t)|cosh k:h-M风y ,z ,R )dhr2+.V ghtanhkh,sir( t√gktanhkh )Ey ,z ,R )Xh(1 )coshkh sinhkhwhere P( x y ,z )is the field point ,Q( xo ,yo ,20 )the source point , g the acceleration of gravity，R the horizontal distance between the field point and the source point , r the distance between the fieldpoint and the source point , and r2 the distance between the field point and the image of the sourcepoint about the seabed ,Ez ,zo,R)= cos{[ k(z + h)]cosl[ K(zo + h )]<( kR),(2)Jo being the zero order Bessel function .Newman( 1990 ) defined the non-dimensional variables X= R/h , Y=zo/h , Z= - z/h , andT=t√g/h with respect to the parameters g and h. With these definitions ,Eq. ( 1 )can be ex-pressed in the formG =gih8(tI F( X,Y- Z)+ F(X ,2- Y- Z)]+ gihz[F(x ,Y- Z,T)+ FX ,2- Y- Z,T)](3)whereF(X ,V)= -; -|e-^ sechkcoshkVJ( kX )Xth(4)√x+V。andF( X ,V,T)=V ktanhk,sin( T V krtanhk )coshVJ( kX )dk .(5)」coshksinhkThus , the Green function can be decomposed to compute the values of Fo( X , V ) andF( X,V,T).2.1 Solution of FoIn order to avoid computing infinite integration , the fllowing form of Fo is adopted ( Newman ,1990 ).When X>1 ,Eq. (4)is easily derived by use of a Fourier series in the resulting expressions asF。= 200s[(nn+中国煤化工(6)m=0here K, is the modified Bessel function of the secMYHCNM HGWhen0≤X≤1 ,Eq. ( 4 )can be approximated by a Chebyshev polynomial. The result takes theform1"°√x+V°Vx +(V+2) x +(V-2)TENG Bin et al ./China Ocean Enginering ,1X3), 2003 ,417- 426+ 1+26. X2"V2n- log2(7)miwhere the cofficient bmm is given in Table 1.Table 1Cofficient bmmm=0m= 1m=2m=3m=40.00000000.01230716- 0.000653410.00003603- 0.00000182- 0.024614280.00522739- 0.000650040.00006286- 0.000003862- 0.001742900.00086822- 0.000195430.00003259- 000005193- 0.0001 14630.00010319- 0.0000387700000882- 000008700.00001243. 0.000004602.2 Solution of FWhen a field point is close to the source point , F ,which includes a singularity , is difficult to becalculated directly. For this reason , Eq. ( 5 ) is decomposed into the formF=F。+[F-F.],.(8)F。and F- F. are defined respectively as :F。= 2|Vksin( TVh )r-2)0( kX )lk(9)0andMV-2) + e-《V+2)F- F。=2|√Vk:J$( k:X )[V tanhksin( T√k tanhh1 - e-4k- sir(下V后)e(1-2) ]1k ,( 10)F。and F- F。are calculated respectively. Thus the value of F can be obtained according to Eqs.(9 )and( 10 ). As there are no singularities involved in F。and F- F。,it is convenient to computethe valueof F。and F- F。. At present there have been many mutual algorithms to evaluate F。thatis the Green function for infinite water depth.However , the direct computation of the above functions is time consuming , and cannot be used ina practical boundary element method , in which million times of calls of the Green functions have to becarried out for a conventional problem. Therefore中国煤化工r the time-domain Greenfunction for finite water depth has to be developecMYHCNM HG3. Polynomial Approximation3.1 Chebyshev Polynomial ApproximationAmong有nomials , the Chebyshev polynomial occupies an important place. From the approx-TENG Bin et al ./China Ocean Enginering ,1X3), 2003 ,417- 426421imation standpoint , the Chebyshev polynomial owns many good numerical properties ,e.g. ,it has theadvantage that it can be truncated to any desired degree of accuracy. And the Chebyshev polynomialhas been widely recognized as a good vehicle to approximate natural signals .For a given function f( x ), it can be approximated by a Chebyshev polynomial expansion( Presset al. , 1986 ):N、1(x)≈[2cT_(x)]-(11)where c(j=1 ,2 .. ,N )is the expansion coefficients of Chebyshev approximation. The sequence ofcoffcients will converge to zero and the series may be truncated to yield any degree of desired accura-cy. The extension to two or three dimensional problems is straightforward.T( x )is a Chebyshev polynomial of degree n ,and is given by the explicit formulaT,( x)= co( narccosx )= > } (( 12)i=0A kind of generalization of the equation , which that is here implemented , is to allow the range ofapproximation to be between two arbitrary limits a and b , instead of just from - 1 to 1. This is affect-ed by change of variables :x-b+a)y=-(b- a)( 13)Thus the coefficient c; of Chebyshev approximation is derived asc; =导乙(x:)];_(x) j= 1,2...，N,( 14)k=0where∈k =[1 k=0l2 k>0.3.2 Ordinary Polynomial ApproximationFor computational convenience , the Chebyshev polynomial is converted to a simpler equivalentform of ordinary polynomial :f(x)=p:x( 15)The ordinary polynomial for a three- variable function ,e.g. F。and F- F。,is expressed by1x,y,z)= 22ZPa( 16)The coeffcient P泳in the above equation cal中国煤化工tion from the Chebyshevexpansion coefficient Ca and the coefficient matrix:TYHC. N M H Gmials:Pa= 222clmahiQmjAnk .( 17)The cofficients of matrix[ a ] are all integers , which are tabulated in books ( Abramowitz andStegun , 1苑方数据。,F- F。and their derivatives are approximated by ordinary polynomials respec-422TENG Bin et al ./China Ocean Enginering ,1X3), 2003 ,417- 426tively in this paper .4. Approximation in Different ZonesBecause the value of the Green function is rapidly oscillatory and its amplitude is large in somezones as shown in Figs. 1~5 ,it is dificult to reach sufficient accuracy if it is interpolated directly inthe whole domain. In this paper the whole domain of independent variables is divided into differentzones according to the contour of each component of the Green function , and the Chebyshev polynomi-als are used to approximate the component of the Green function in each zone .0.20.2 1510.10)-0.1.-0.216-21.5X=1012T =100.5k4r1000Fig. 1. Contourof F2( X=10).Fig. 2. Contourof F=( T= 10).0. 44V=00.40.8V=10.310.210. 1-0.1-0. 1-0.4-0. 1-155071;“ xT0个0Fig. 3. Contourof Fa( V=0).Fig. 4. Contourof F2( V=1).5吉5青v=2hf 155手-25.5^中国煤化工TYHCNM HGFig. 5. Contourof F=( V=2).As sh有教据igs. 1 ,2 ,and5 ,F。changes rapidly near V=2. From Figs. 3 and 4 ,it can beTENG Bin et al ./China Ocean Enginering ,1X3), 2003 ,417- 426423seen that F。changes rapidly near V=0 ,1 where X and T are small. .The contour of F- F。is relatively smooth ,and so it is not necessary to divide the domain of in-dependent variables. W hen higher accurate results are required , the valueof L ,M or N must be in-creased. At the same time , the burden of computation will increase with the increase of expansionterms. In this case it may be more appropriate to subdivide the domain of approximation to acceleratecomputation rather than increase the value of L ,M or N.5. Numerical ResultsSeveral examples of F - Fx and its derivatives have been given to show the utility of rmultivariatepolynomial approximations in the evaluation of the free-surface Green function. Comparisons are madebetween the results from direct computation and approximation. Tables 2~ 5 are the comparisons of anapproximation in the zoneof x∈[2 ,3], v∈[0,2], T∈[3 ,4]with L= M= N= 10 and the di-rect computationof F- F. and its derivatives with respect to X , V and T at V=1.2and T=3. Itcan be seen from the tables that the results from the approximation are very close to the directly com-puted values when the number of truncated terms is only 10. More accurate results can be achievedwith the increase of the number of truncated terms , but at the cost of computing time. In practicalhydrodynamic computation , the present results are accurate enough in the engineering sense .Table 2Valuesof F- F。Actual valueChebyshev fitPolynomial fitRelative error2.00.01 8997500.018998405. 17688E-052.10.005825720.005826130.005826167.61752E-052.2- 0.0065062- 0.00650542- 0.006505391.23963E-042.3- 0.0179415- 0.01794103.15605E-052.4- 0.0284415- 0.02844191.15918E-052.5- 0.0379843- 0.03798522. 33417E-052.6- 0.0465639- 0.0465644- 0.04656438.48040E-062.7- 0.0541890- 0.0541887- 0.05418868.04331E-062.8- 0.0608810- 0.0608809_ - 0. 06088093.05949E-07中国煤化工2.9- 0.0666722- 0.06667.58747E-06MHCNMHG3.0- 0.07160434.16209E-07TENG Bin et al ./China Ocean Enginering ,1X3), 2003 ,417- 426Table 3Valuesofζ F- F。]aXXActual valueChebyshev fitPolynomial fitRelative error2.00. 13568300.13568301.42771E-062.10. 12762800.12762807.00529E-072.20.11891600. 11891608.77161E-072.30. 10972706.79011E-072.40. 10023600. 10023707.43301 E-072.50.09060930.09060949.04503E-072.60.08099815.51908E-072.70.07153960.07153970.07153981. 77048E-062.80.06235360.06235371.43387E-062.90.05354070.05354081.32199E-063.00.04518231.07185E-06Table 4Valuesof[ F- F. ]aV- 0.0812526- 0.0812528- 0.08125271.00866E-06- 0.0902984- 0.09029851.65021E-06- 0.0977732- 0.09777333.81013E-07- 0.10368501.36530E-06- 0.10807301.03411E-06-0. 110020- 0.110020- 0. 1100207.38332E-07- 0.11256209. 92866E-07- 0.11285906. 60167E-07- 0.1120160- 0.11201中国煤化工.64676E-07- 0.1101630- 0.110161:AYHCNMH(.76321E-07- 0.10743806.24131E-07TENG Bin et al ./China Ocean Enginering ,1X3), 2003 ,417- 426425Table 5Valuesof[ F- F。]aTXActual valueChebyshev fitPolynomial fitRelative error2.00. 18494000. 18493800.18493701. 12802E-052.10.17256700.17256400. 17256401. 65792E-052.20. 15984300. 15984800.15984802.88993E-052.30. 14693400. 14693601.54149E-052.40. 13397600. 13397700.13397704.67135E-062.50.121 12400. 12112400.12112401. 16872E-062.60.10851400.10851600. 10851601.33200E-052.70. 09626620.09626718. 90049E-062.80.08448260.08448210.08448224.76230E-062.90.07324680.07324748. 44267E-063.00.06262270.06262300.06262316. 54367E-066. ConclusionFor solving large amplitude non- harmonic motions of ships , a numerical scheme is implementedfor fast evaluation of the 3D time-domain free surface Green function for finite water depth and its asso-ciate derivatives are obtained in the research. Polynomials are used to approximate the Green functionto a sufficient accuracy in a simple and computationally efficient manner. Formulas of multi-dimension-al Chebyshev polynomial are given to avoid the time consuming direct integration computation. For fur-ther acceleration of the computation , the Chebyshev polynomial can be converted into an ordinary poly-nomial. Thus the task of evaluating the Green function and its associate derivates is greatly simplifiedcompared with the direct numerical integration. The comparison of the approximation method with thedirect method for the Green function and its derivatives shows that the approximation algorithm canyield a very high accuracy with not too many polynomial terms . 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