the introduction of a criterion used to truncate the influence of the Green function and its derivatives. The other ...... H. P. Wilham, A. T. Saul, T. V. William and P. F..
Pergamont PII: SOO45-7949(96)0026&l
compu’ers & .smKrures Vol. 62, No 4. pp ~3-610. 1997 Copyright ~Q 1996 Elsewer Saence Ltd Pnnted I” Great Bntam All nghts reserved w45-7949/97 5 I7 00 + 0.00
COMIPUTATIONALLY EFFICIENT TECHNIQUES IN THE HYDROELASTICITY ANALYSIS OF VERY LARGE FLOATING STRUCTURES Suqin Wang,? R. C. Ertekintt and H. R. Riggs8 t Department of Ocean Engineering and 0 Department of Civil Engineering, University of Hawaii at Manoa, Holmes Hall 402, 2540 Dole Street, Honolulu, HI 96822, U.S.A. (Received 3 I May 1995) Abstract-Two techniques are introduced in the three-dimensional hydroelastlcity theory to increase the computational efficiency for the determination of the dynamic response of very large floating structures (VLFS). One technique is related to the convergence of the Green function and its derivatives, namely the introduction of a criterion used to truncate the influence of the Green function and its derivatives. The other involves using an iterative sparse solver for the linear system of equations. The principle motivation behind the application of these two techniques stems from the fact that a source makes a very small contribution to the potential at a point “far away” from the source point. By employing these two techniques in the hydroelastic analysis of a VLFS, the CPU time and required storage are considerably reduced, and therefore it is now possible to analyze the dynamics of a VLFS as large as a floating airport by using the three-dimensional panel method. Copyright 0 1996 Elsevier Science Ltd.
1. INTRODUCTION large floating structures (VLFS) have been proposed for many applications such as floating industrial plants, cities and airports. Because of the enormous size of these proposed structures, e.g. perhaps 4000 m long for an airport, the flexible modes of motion are important such that the interaction between the motions of flexible modes and waves must be included in the dynamic response analysis using a proper hydroelasticity theory. The three-dimensional hydroelasticity theory, which utilizes structural mode shapes and the source-distribution method of potential theory for the hydrodynamic analysis, is the most complete unified theory to date for obtaining the dynamic behavior of deformable bodies within the assumption of linearity. However, limited computer resources make the application of the theory to a VLFS very difficult. This difficulty is mainly related to the hydrodynamic analysis. In the hydrodynamic analysis, the wetted surface of the body is represented by a large number of discrete panel elements. The potentials at the panels are described by the corresponding linear system of equations. This linear system of equations is characterized by a fully-populated square matrix of complex coefficients with its dimension equal to the number of panels. The coefficients are related to the source potential and its derivatives. Therefore, the difficulty stems from two numerical problems: one is Very
$ To whom correspondence should be addressed.
the evaluation of the complex coefficients and the other is the solution of the linear system of equations. To overcome the difficulty, Newman [I] has developed an efficient numerical algorithm to calculate the source potential and its derivatives to reduce the computational time. Wu [2] introduced the composite source distribution method to increase the computational efficiency in solving the linear system of equations when the body is of single symmetry. The method was later extended to a double symmetric body [3]. However, the hydroelastic analysis of a VLFS still remains untractable even with the use of these techniques if the structure size is on the order of an airport. To this end, some other efficient hydroelasticity methods have been developed (see for example, Ertekin [4] for a review). However, they are applicable to specific structural geometries. In this paper, an efficient approach that is applicable to any kind of structural geometry is developed. The approach includes not only the very efficient double-composite source-distribution technique mentioned above, but also two other new techniques introduced for the first time in a hydroelastic analysis. The first technique is related to the computation of the potential coefficients, which relate the velocity potential to source strengths, and the influence coefficients, which relate the normal derivatives of the velocity potential to source strengths. We use the fact that the potential and influence coefficients are very small when the source and the field points are far away from each other to establish a criterion to determine under what conditions they can be 603
Suqin Wang et al.
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assumed to vanish. This criterion is then used tq calculate only the large, most significant, coefficients, thereby reducing the CPU time. The other technique involves the solution of the system of equations that relate the known boundary conditions (normal derivatives of the potential) to the calculated influence coefficients and the unknown source strengths. Once the influence (and the potential) coefficient matrix is calculated by using the criterion mentioned above, many elements of this matrix will have zero entries, i.e. it will be a sparse matrix which is diagonally dominant. Such a matrix can be solved efficiently by an iterative sparse solver. We develop an algorithm which is based on the diagonal-preconditioned conjugate gradient method (see for example Hackbusch [5]) to solve these complex equations. We will apply the new techniques developed here to the solution of a hydroelasticity problem which involves a VLFS. The computational efficiency that we achieve by using these techniques will be illustrated by examples, paying attention especially to the reduction in the CPU time. At the same time, the inaccuracies that are caused by the implementation of the criterion for zero entries of the coefficient matrices will be discussed. 2.
EQUATIONS OF MOTION
The motions of the body and the fluid are described in this paper in a Cartesian, right-handed coordinate system Osyz, with the Oz axis pointing upward. If the body is discretized such that it has N displacement degrees-of-freedom, then the displacement, u, at any point on the body can be expressed as: (U(X,Y, z, t)} = i ($M.~, L’, z)}pr(t), ,=I
(1)
where {$&x, y, z)} denotes the rth principal mode shape vector of the dry structure and p,(r) is the corresponding principal coordinate. It is assumed that the fluid is incompressible and inviscid, the flow is irrotational, and the wave amplitude is relatively small. Hence, the fluid motion and the hydrodynamic actions around a flexible body can be determined by employing the three-dimensional potential theory [6]. It is further assumed that the body is excited by a train of regular, long-crested waves and thus the responses are harmonic with a frequency w, i.e. pr(t) = pr e.-“‘“.
(2)
The generalized linear equations of motion of the floating body in waves is then expressed as [7]:
[ - o’([M?] + [M?]) - io([C?l + KY]) +
([PI + W,l{P) = {G),
(3)
where {p} is the vector of principle coordinates, [Ml, [C$] and [K$] are the diagonal modal mass, damping and stiffness matrices of the structure, respectively; [M:] and [C:] are the generalized added mass and damping coefficient matrices, respectively; [K:] is the generalized hydrostatic restoring matrix; and {F8) is the vector of generalized wave exciting forces. The generalized added mass and damping coefficient matrices and wave exciting forces are obtained from the hydrodynamic analysis. Note that matrices with subscript f in eqn (3) are not diagonal. For eqn (1) to be valid, it is not necessary that the fluid-related matrices be diagonalized by the principle modes [8]. The present method of solution is not the standard mode superposition method, which relies on the uncoupling of the equations of motion in principle coordinates. Rather, in hydroelasticity, a truncated set of principle coordinates is used to reduce the dimensions of the system of equations of motion. The truncation is possible because the principle coordinates are more efficient in representing the structural motion than are the “physical” coordinates. 3. EFFICIENT TECHNIQUES IN THE POTENTIAL ANALYSIS
The velocity potentials are usually solved using the Green function method. The velocity potential 4(x, y, z) at any point is expressed as:
‘#+> J’, z) =
G(x, I’, z, 5, q, [)a((,
‘I, [) dS, (4)
where G(.u,y, z, 5, q, [) is the Green function, ~(4, II, [) is the source strength, (x, y, Z) and (, z,, t, ?, 0 ds,
;~G(x,.y,,z,,5,il,i)ds. A,.
It can be seen th.at two distinct numerical problems are involved in solving the source strengths and velocity potentials, i.e. eqns (6) and (7), for a very large floating body. The first one is the evaluation of the matrix of source potentials [a], and derivatives of source potentials [/I]. These are complicated mathematical functions (see eqn (12)) and need to be evaluated for each Icombination of panels. The second one is to solve the linear system of equations i.e. eqn (7) for the source strengths. The solutions of these two numerical problems are regarded as the main computational difficulties in performing the three-dimensional hydroelastic analysis of a VLFS. To overcome these difficulties, we now introduce two techniques. One is related to the evaluation of the matrices and the other is related to the solution of the linear system of ecluations.
605
of the first kind of order zero, and PV indicates the principal value of the integral. We employ the integral form of the Green function rather than the series form because we are interested in an algorithm which is especially efficient when kR is not very large, as discussed in Ertekin et al. [lo]. Figure 1 shows how the Green function and its normal derivatives vary with kR. It can be seen that the Green function and its normal derivative, both of which are oscillatory, decrease sharply as kR increases. Hence, when the distance between the source and field point is large, the corresponding coefficients a,, and /J, will be sufficiently small such that they can be assumed to vanish. Based on this property of the Green function, we choose a critical value, say y, to determine when the coefficients can be assumed to be zero. Specifically, when kR > y then the coefficients are set to zero. For a VLFS, many coefficients can be set to zero by using this criterion, hence significant savings in CPU time spent on the calculation of the coefficients can be achieved. It is clear that the Green function is dependent on kR and not specific body geometry. However, when one compares kR to the chosen Green function cut-off criterion, y, then it is also clear that the choice of y needs to somehow involve the body geometry. 3.2. Iterative sparse solver
3.1. Calculation of the matrices The Green function given by Wehausen Laitone [9] is expressed as
and
G(x, y, z) = f + G*,
(10)
R = [(x - r)* + (y - q)’ + (z - r/)2]“‘,
(11)
The matrices [a] and [j?] that are obtained using the criterion mentioned above are large, sparse matrices
where Real part _.____--.-.,maginary pau
r (p + v) e-PAcosh[p(< + h)]
G*=++2PV
,u
2
sinh(ph) - v cosh(gh)
* cosh[p(z + h)]J&r)
dv
+ i 2n(k2 - v2)cosh[k( 5 + h)] k*h - v2h + v
* cosh[k(z + h)]Jo(kr),
(12)
in which R2 = [(x -
0’ + ~(y - rj)2 + (z + 2h + [)2]“2, r = [(x -
’ ,’
i.o-
5
6
7
8
9
-. -.._
-v
10
-.
‘-...
11
12 Period (sec.)
Fig.
2.
Displacements
at
the
bow
of
the
box
(80 x IO x IOm). 4. CPU SAVINGS BY THE PRESENT METHOD IN HYDRODYNAMIC ANALYSIS OF A VLFS
Both techniques, namely the cut-off criterion for the Green function and its derivatives and the sparse solver, introduced in the present method contribute to CPU savings, and the sparse solver also contributes to savings in storage. We next discuss these savings in detail. 4.1. CPU savings by using the cut-08
criterion
A box with the dimensions of 500 x 60 x 20 m and 10 m in draft is chosen in this particular study. The main particulars of the box are given in Table I. The whole wetted surface is discretized mto 1872 panels. Since y is related to both the distance between the source and field points as well as the wavenumber, the CPU savings are different for each wave period if the same criterion is used. Table 2 lists the CPU time required for the calculation of the Green function and its derrvatives for three wave periods and nine different criteria. As expected, Table 2 shows that the smaller the influence distance determined from 7 = kR is, the more CPU savings can be achieved. Figure 3 shows the results of the responses when different criteria are used for wave periods ranging from 8 to 26 s. The responses shown are the heave and pitch motions and principle coordinates of the first and second bending modes. The wave heading IS zero degree. Figure 3 shows that the predicted responses are close when ;I > 4.
Table
I. Mam
particulars
of the box
Length, wrdth, depth (m) Draft (m) Displacement (kg) Mass moment of inertia I,, (kg m’) I, (kg m’)
500 x 60 x 20 IO 3.075 x IO” 1.025 x IO” 6.416 x IO”
I_, (kg m’) Modulus of elasticity, E (N m -‘) Cross sectlonal rigidity, EI (N ml)
6.498 x IO’? 2.11 x IO” 2.954 x lOi
607
Hydroelasticity analysis of very large floating structures Table 2. CPU time in the calculation of the Green function
large error in the principle coordinate of the second-bending mode when using y = 10. We have not found a good explanation for this yet. We note 0.6 ;) 15 12 10 8 : 4 2 I that, in strip theory, the interference of the source and 98 78 58 36 28 24 8 146 126 1I I field points are considered only within one cross 12 193 191 183 166 138 103 62 37 29 section. In the present method, if y is chosen such that 16 202 192 190 185 176 143 91 55 40 the interference distance is larger than the maximum distance within any cross section, the results obtained are at least better than that obtained by strip theory. The relative errors are different at each wave period Figure 4 shows the relative errors of the responses for the same y (see Fig. 3). For example, while y = 4 corresponding to each criterion for the wave period is adequate for one wave period, it may not be of 8 s. The “exact” responses are obtained using y = 35 such that all the elements in the matrices [do] adequate for another wave period. More detailed studies about the criterion are necessary to provide and [fl] are included in the calculations. The general guidance for the user to choose the criterion in the trend of the relative errors decrease as y increases calculation. although the error,s oscillate within a certain range. If y = 4 is chosen as the critical criterion in the Quantitatively, the relative error should decrease as calculation of the Green function and its derivatives, y increases. The oscillation of the errors may be due namely the responses obtained using that criterion to the oscillation ‘of the Green function itself and are reasonable, then for an 8-s wave, the CPU time error cancellation for a specific y . There is a relatively and its derivatives (s)
Y
15
___
Y
~
15
0.006
8
10
12
14
16
18
20
22
24
26
Period (sec.)
Period (sec.)
S(a) Heave response
3-(b) Pitch response
5
Y
15 12 10 8 6 4
Y 15
.-s E:
0.05.
12
2 B5
0.04-
. .
10 8
: 1
0.03 -
. .
6 4
~
2
Period (sec.)
Period (sec.) 3-(c) Response of the first-bending mode
3-(d) Response of the second-bending
mode
Fig. 3. Responses of the box using different criteria (500 x 60 x 20 m). (a) Heave response. (b) Pitch response. (c) Response of the first-bending mode. (d) Response of the second-bending mode.
Suqin Wang et al.
608
100 f
in which MT is the number of iterations required for convergence. If the matrix [A] is sparse, the number of operations can be reduced by using an iterative sparse solver based on the diagonal preconditioned conjugate gradient method. If the number of nonzero elements of matrix [A] is NS, the number of operations required to solve the equations is
I - Heave
60
--+----+--
Pitch 1-banding mode
---+--
2-bending mode NOP = MT * [2NS + O(N)].
1 2 3 4
5
6 7 8 9 IO 11 12 13 14 15 Criterion y
Fig. 4. Relative errors of the responses (period = 8 s)
used for the calculation of the Green function and its derivatives is reduced from about 200 s, which includes calculation of all the coefficients, to 58 s, that is about one-quarter of the original CPU time. This CPU saving corresponds to the case of the given box which has a total of 1872 panels. For a larger floating body, more panels will be required, and therefore the CPU time will be reduced even more if the criterion is applied. Thus, the larger the floating body, the more CPU time savings will be achieved by using the criterion proposed here. 4.2.
CPU
savings using the iterative sparse solver
Equation (7) is usually solved using a direct method that terminates after many operations with an exact solution (excluding the round-off errors). For a general unsymmetric matrix, for example the coefficient matrix in eqn (7) direct solution of a system of N equations [A]{x} = 16) requires the following number of operations by using Gaussian elimination method: NOP = 2N’/3
+ O(NZ),
(16)
where N is the total number of equations and NOP is the total number of operations, in which each addition, subtraction, multiplication or division is counted as one operation. For the iterative solution of a system of equations, one starts with an arbitrary initial vector {x0} and computes a sequence of iterates Ix,.} for m = I, 2, . . Any iteration requires at least the computation of [A]{x,,,) - {b). For a general N x N-matrix [A], the multiplication [A]{x,,,} would require 2N’ operations. The diagonal preconditioned conjugate gradient method for the solution of the system of equations requires the following number of operations: NOP = MT * [2N? + O(N)],
(17)
(18)
If NS is much smaller than N x N, the CPU time in solving eqn (7) can be reduced considerably. The box defined above is used again as a model to study the CPU savings in solving eqn (7). The single composite source distribution method [2] is employed for the solution of the potentials. In the single composite source distribution method, the two composite source strengths are calculated. Therefore, eqn (7) is replaced by two linear systems of equations, each one involving N/2 equations. In other words, in the single source distribution method two linear systems of N/2 equations are solved to obtain the composite source strengths. The CPU savings in solving the linear system of equations are studied from the viewpoint of the convergence of the method and the sparsity of the matrix. Three discretizations for the box are used. The total number of panels (N) are 524, 1098 and 1872. The results of the motion responses obtained using these three discretizations are found to be close to each other. Figure 5 shows the number of iterations required for different wave periods and the motion responses for the case of 1872 panels. The tolerance used is 1O-5. It can be seen that the number of iterations required to obtain the converged results is much smaller than the number of panels. Comparison of eqns (16) and (17) shows therefore that the number of operations required reduces significantly if the iterative method is used. Figure 5 also shows that
4Or
--__ Heave ..~ .._.. _. pitch
-~~------~ DIffraction
lot,,,,‘,,,,‘,,,,‘,,,,‘,,,,‘~,,,’,,,,’,,,,’,,,,l
8
10
12
14
16
18
20
22
24
26
Period (sec.)
Fig, 5. Number of iterations m solvmg eqn (7) (total number of panels is 1872).
Hydroelasticity
50
r -.
b .P
structures
609
iterative sparse solver is therefore about 22% of the CPU time required using the iterative solver.
12sec.
---t_
16sac.
5. DISCUSSION AND CONCLUSIONS
20 sac. 24sec.
i -
1100
800 Fig. 6. Number
of very large floating
esec.
__c_40
analysis
1400
1700 Total panel number
in solving eqn (7) for diffraction source strengths.
of rterattons
the number of iterations required decreases for the long wave periods. This is because a better initial guess can be made based on the results for the previous period. Figure 6 presents the number of iterations required in solving for the diffraction source strengths for three discretizations. The same tolerance is used. It can be seen that the number of iterations slightly increases for the larger panel case. According to eqns (16) and (I7), the iterative method can provide a big CPU savings compared with the direct method, especially for the analysis of a VLFS. We now estimate the CPU time required to solve eqn (7) by using Gaussian elimination and the iterative method for the case of 1872 panels. The dimension of the linear system is 936 by using the single composile source distribution method. If we solve eqn (7) for the diffraction source strengths for an 8-s wave, the approximate number of operations required is about 5.47 x 10” by using the Gaussian elimination method, and 5.43 x IO’ by using the iterative method. Therefore, the CPU time required using the iterative method is about one-tenth of the CPU time using the Gaussian elimination method. By using the iterative sparse solver, more CPU time can be saved. If the criterion y = 4 is used in the calcufation of the matrix [p], for an 8-s wave, the influence distance determined from the criterion is about 64 m. For the box, about 78% of the matrix elements are zero. From eqns (17) and (18), it can be seen that the CPU time in solving the equations is much less by using the iterative sparse solver than using the iterative method. To compare the CPU time required by using the iterative solver and the iterative sparse solver, we choose, as an example, the case when y = 4 and the wave period is 8 s. From eqns (17) and (18) the ratio of the number of operations using the iterative solver .and the iterative sparse solver is about N’/NS. The CPU time required using the
Two techniques are introduced for the hydroelastic analysis of a VLFS to increase the computational efficiency. By using a cut-off criterion for the Green function and its derivatives, the CPU time for the calculation of the Green function and its derivatives is reduced, and sparse matrices instead of full matrices result. The iterative sparse solver is developed and applied to the linear system of equations that is characterized by a sparse matrix of complex coefficients to further reduce the CPU time. By comparing the results obtained using different criteria, it is shown that an appropriate criterion can be found for all practical purposes. The CPU time is reduced significantly by applying the criterion and using the iterative sparse solver. For the 500 m long box used, the reasonable criterion is about 6. It is also shown that the relative errors due to the use of the criterion are different at each wave period. A more detailed study to find a general guidance to choose the criterion is needed. Work in this direction is in progress. In this study, the same criterion is used for the convergence of the Green function and its derivatives. From Fig. I, the derivatives of the Green function converges faster than the Green function itself, namely the influence of the derivatives of the Green function vanishes within a shorter distance than that of the Green function. More error is caused by the truncation of the Green function than the truncation of its derivatives when the same criterion is used. It may be more reasonable to use separate criteria for the Green function and its derivatives. This is also being studied currently. By using the iterative sparse solver, the computational storage requirement is also reduced, since only the nonzero elements need to be stored. These two techniques should be especially useful for the hydroelastic analysis of a VLFS which usually causes computational difficulties. Acknowledgemenrs-The authors would hke to thank Dr C. J. Reddy, NASA-Langley Research Center, for providing the iterative solver for the full matrix case, which was used to develop the sparse solver. This work is supported by the U.S. Nattonal Sctence Foundation under grants BES-9200655 and BCS-8958346. SOEST contribution no. 4121.
REFERENCES 1. J. N. Newman, Algorithms for the free-surface function. J. Engng Ma/h. 19, 57-67 (1985).
Green
2. Y. S. Wu, Hydroelasticity of floating bodies. Ph.D. Dissertation, Brunei University, U.K. (1984). 3. Y. S. Wu, D. Wang, H. R. Riggs and R. C. Ertekin, Composite singularity distrtbution method with appli-
610
4.
5. 6.
7.
Suqin Wang et al. cation to hydroelasticity. Marine Strucr. 6, 143-163 (1993). R. C. Ertekin. Current and future directions in very large floating structure research and development. In: Proc. Techno-Ocean ‘94 Int. Symp., 26-29 October, Kobe, Japan, Volume on Invited Lectures, pp. 23-29 (1994). W. Hackbusch, Iterative Solution of Large Spare Systems of Equations. Springer, Berlin (1994). R. E. D. Bishop, W. Cl. Price and Y. Wu, A general linear hydroelasticity theory of floating structures moving in a seaway. Phil. Trans. R. Sot. Lond. A 316, 375-426 (1986). S. Wang, R. C. Ertekin, A. T. F. M. V. Stiphout and P. G. P. Ferier, Hydroelastic-response analysis of a box-like floating airport of shallow draft. In Proc. 5th
Int. Ofihore and Polar Engineering Conj., The Hague, The Netherlands, I l-16 June (1995). 8. R. W. Clough and J. Penzien, Dynamics qf Structures. McGraw-Hill, New York (1975). 9. J. V. Wehausen and E. V. Laitone, Surface waves. In: Handbuch der Physik (Edited by S. Flugge), Vol. 9. pp. 446-776. Springer, Berlin (1960). -IO. R. C. Ertekin. H R. Rians. X. L. Che and S. X. Du, Efficient methods for- hydroelastic analysis of very large floating structures. J. Ship Res. 31, 58-76 (1993).
Il. H. P. Wilham, A. T. Saul, T. V. William and P. F. Brian, Numerical Recipes in Fortran. Cambridge University Press, Cambridge (1992). 12. J. N. Newman, Wave effects on deformable bodies. Appl. Ocean Res. 16, 47-59 (1994).
