Efficient resolution of linear systems involving in wave radiation and diffraction around multiple bodies

Transcription

1 Proceedings of the Seventh International Workshop on Ship Hydrodynamics, September 16-19, 211, Shanghai, China Efficient resolution of linear systems involving in wave radiation and diffraction around multiple bodies Y. Su 1, W.Y. Duan 1 & X.B. Chen 2,3 1 Shipbuilding Engineering College, HEU, 151 Harbin (China) 2 Research Department, BV, 9257 Neuilly-Sur-Seine (France) 3 Professorship, HEU, 151 Harbin (China) Abstract: Boundary element methods to evaluate the wave radiation and diffraction around bodies involve a large linear system which is complex, full, non diagonal-dominant and of large size. In order to solve it efficiently, the algorithm based on the generalized minimal residual algorithm (GMRES) and incomplete LU precondition (ILU) has been developed. A parameter µ is defined to quantify the incompleteness of LU decomposition. The optimizations of this ILU-GMRES algorithm by varying the value of µ is then studied by numerical tests of bodies with different geometry. It is shown that a significant reduction of CPU time can be obtained even for complex offshore structures. Key words: Incomplete LU factorization, GMRES, Boundary element method, Offshore structure, Optimization 1 Introduction Efficient resolution of the linear system Ax = b involving in the problem of wave radiation and diffraction around bodies by using the classical boundary element method, is critically important since the system matrix is complex, full, non diagonal-dominant and of large size. Indeed, the resolution of linear systems costs the major part of CPU expense in routine applications of seakeeping computations. The usual direct methods like Gauss Elimination or LU decomposition are not practical, especially, in the analysis of interactions between multiple bodies. The generalized minimal residual algorithm (GMRES), as an iterative (indirect) method, has been used to solve this type of problems combining with some preconditioners to reduce the number of iterations. For different numerical models, it is important for us to select a proper preconditioner for the iterative method, and the optimal selection between the direct method and the iterative method. The GMRES approach was first proposed by Saad and Schultz (1986), which is used for nonsymmetric and not necessarily positive definite matrices. Then the solver was tested in the fluid dynamics context for three dimensional models of viscous flows or boundary element analysis of structural problems by Kane et al. (1991) and Habashi et al. (1992). Kane et al. concluded that the GMRES solver is faster than direct solvers for a given large matrix. And it is more consistent than other iterative solvers due to convergence characteristics as shown in Habashi et al. (1992). By an out-of-core treatment, the classical RAM requirement in the solution of boundary element analysis of large offshore structures has been reduced to O(n) instead of O(n 2 ) in Y. Zhao and J.M.R. Graham (1996). The modified version of the incomplete LU factorization (ILU) was used as a preconditioner for GMRES in hydrodynamic analysis of the interaction between the global structural response of a very large floating structure and the associated diffraction wave field in an unbounded exterior domain by Noritoshi Makihata et al. (25). As for the boundary element solution of offshore structures, the constant element method based on frequency-domain free surface Green function has been widely used to solve the wave radiation and diffraction problem of offshore structures. There have been some hydrodynamic software based on this method, like HydroStar developed by Bureau Veritas (France). It also consumes a large amount of CPU time and memory requirement for large and complicated offshore structures if the direct method is used. In order to solve the problem, the GMRES approach combining with a preconditioner was tested. In this paper, firstly, the effectiveness of the GMRES program with the preconditioner ILU is tested

2 for the solution of linear systems involving in boundary element method of wave diffraction-radiation around floating bodies. Secondly, in order to improve the efficiency, the value of the parameter µ in the preconditioner varies according to different shapes of floating bodies. Thirdly, as for different linear systems, a judgment formula has been presented to select more time-saving method between the iterative method and direct method. 2 ILU-GMRES algorithm The GMRES approach was proposed for solving nonsymmetric linear systems in Saad and Schultz (1986). First we construct an orthonormal basis by using Arnoldi process. Then the norm of the residual vector over a Krylov subspace is minimized at every step. If the coefficient matrix A C n n is nonsingular, b C n, and x C n is an initial guess. After k steps of Arnoldi iteration we have the factorization following Golub and Van Loan (1996): AQ k = Q k+1 Hk (1) in which the columns of Q k+1 = [Q k q k+1 ] are the orthonormal Arnoldi vectors and H k is a (k +1) k matrix which has the form : [ ] H H k = k (2),,,h k+1,k in which H k is an upper Hessenberg matrix H k = Q H k AQ k of order k where Q H k is the conjugated transpose of Q k. In the kth step of GMRES, the norm b Ax k 2 is minimized subject to the constraint that x k has the form x k = x + Q k y k for some y k C k. If q 1 = r /ρ in which ρ = r 2, then it follows that: b A(x +Q k y k ) 2 = r AQ k y k 2 = r Q k+1 Hk y k 2 = ρ e 1 H k y k 2 (3) in which e 1 is the first column of (k+1) (k+1) identity matrix. Thus, y k is the solution to a (k+1)- by-k least squares problem and that can be efficiently solved using Givens rotations. The solution of the linear system is given by x k = x +Q k y k. In order to acquire precise numerical solution, backward error analysis is used for stop criterion. As in Dai (23), the stop criterion adopted in the program is: ǫ(x k ) = b Ax k 2 / b 2 (4) by the ratios of the residue vectors with respect to right hand side. If ǫ is less than a parameter (1 5 ), the program will stop. At high wave frequencies, the number of iterations will increase sharply due to ill-conditioned coefficient matrices. Preconditioning is one of the key to making GMRES effective. The incomplete LU factorization is one of the most effective preconditioner according to Quarteroni et al. (2). After multiplying the ILU on both sides of Ax = b, the coefficient matrix tends to the identity and the right hand sides will tend to the solution. As for the ILU program, the process of decomposition is the same as the classical LU factorization but only part of the elements can be decomposed. We set a parameter µ to control the elements which should be decomposed. a(i,j)/a(i,i) > µ for j = i+1,i+2,,n (5) For the ith row, the elements which satisfy the above equation will be decomposed. The value of the parameter µ varies due to different linear systems, which will be presented in the following section. The algorithm of ILU is sketched below : for k = 1 : n-1 A(k+1 : n, k) = A(k+1 : n, k) / A(k, k) for i = k+1 : n if ( formula(5) ) then for j = k+1 : n A(i, j) = A(i, j) - A(i, k) * A(k, j) end endif end end A case of cylinder with a radius of R = 12.5m and draft D = 3R was tested. As for comparison, the direct solver based on the subroutine ZGESV from the LAPACK library. ZGESV computes the solution

3 Table 1: Comparisons of solutions by using LU and ILU-GMRES methods RhS M j (LU) E j (LU - ILU-GMRES) D D D+.2551D D D D D D D D D D D 6 to a complex system of linear equations Ax = b using LU factorization with partial pivoting, where A is an N N matrix and x and b are N M matrices with M the number of second members. Firstly, the ILU-GMRES solver was validated by the cylinder with 318 panels on the surface of the body. The nondimansionalfrequencyofanincidentwaveisω 2R/g =.6rad/s. Thefirstsixcolumnsofright-hand sides in the linear systems represent radiation components and the last column represents diffraction. In Table 1, RhS means the right-hand sides of the linear system. M j means the average of absolute value of all the elements in each column of the solution obtained by the direct method. M j = N x(i,j) /N for j = 1,2,,7 (6) i=1 Compared to the results by the direct method, we apply the following equation to compute the relative error. N E j = x (i,j) x(i,j) /N for j = 1,2,,7 (7) i=1 In Table 1, the first column represents the column number of the right hand side matrix. The second column represents the average of the solution of linear system solved by the direct method. Compared to the solution of the direct solver using LU decomposition, the third column presents the relative error between the direct method (LU) and the preconditioned iterative method (ILU-GMRES). We can find the solutions of linear system solved by ILU-GMRES are very good. That means the program GMRES with preconditioner ILU can provide precise solution for the linear system arising from the boundary element analysis of wave radiation and diffraction around offshore structures. Based on the former numerical model cylinder, we tested a set of wave frequencies. The CPU time of different methods has been depicted against nondimensional wave frequencies on Figure 1. It is shows that the GMRES method is faster than LU factorization for this numerical model. But it will cost more and more time at higher frequencies due to the increase of iterations. Using the ILU preconditioner, the CPU-TIME at high frequencies is reduced significantly as the number of iterations is decreased due to the preconditioner CPU_TIME(s) 15 1 direct method GMRES ILU_GMRES Figure 1: Mesh of the cylinder (left) and CPU time for different methods (right)

4 The second example concerns the case of multiple bodies. Two LNGs of different size (L=275m for the larger one) and with different draft are in the position of side-by-side as shown on the left part of Figure 2. The mesh composed of 3134 panels ( ) for both hulls is used in the computation. Both the direct method (LU decomposition) and the iterative method (ILU-GMRES) have been used to solve the linear system. The comparison of CPU time by using both methods is depicted on the right part of Figure 2. It is shown that ILU-GMRES can save at least half of CPU time with the same accuracy direct method ILU_GMRES CPU_TIME(s) Figure 2: Mesh of two LNGs (left) and comparison of CPU time (right) 3 Optimizations The method based on extending integral equations to the interior waterplane has been widely used to remove the irregular frequencies. Panels are then distributed on the interior free surface of a ship or an offshore structure. The surface of interior waterplane for a ship is much larger than that for a semisubmersible, TLP or a spar. Four models of different geometry like cylinder, TLP, 2LNGs and container ship will be tested. The cylinder has its radius and height equal to 12.5m and 37.5m, respectively. The TLP presented in Eatock Taylor and Jeffrey (1985) was used. We also test the classical ITTC S-175 container ship. The surfaces of above 4 structure are represented by the meshes composed of 3532, 3432, 3134 and 3422 panels for the cylinder, ITTC TLP, 2LNGs and containership S175, respectively. A series of wave frequencies has been used with their non-dimensional values varying up to 12. To note that the non-dimensional frequency is defined as ω L/g with wave frequency ω associated with the characteristic length L taking as 1R, 13m, 275m and 175m for the cylinder, TLP, 2LNGs and S175, respectively TLP S175 2LNG Cylinder The number of iteration The number of iteration Figure 3: Numbers of iterations for different frequencies (left) and for different values of µ (right) On the left part of Figure 3, the CPU time is presented for different structures associated with a large range of wave frequencies. The CPU cost for 2LNGs and S175 are much more than cylindrical structures. As we know, the parameter µ represents how complete is the LU decomposition. If we reduce the value of the parameter, the number of iteration can be decreased. We use the case of 2LNGs for our experiments.

5 According to the right part of Figure 3, we can see the number of iterations can be reduced through declining the value of µ. But the CPU-TIME does not always decline along with the number of iteration due to the incomplete LU decomposition. It will cost more CPU-TIME when we choose small value of µ. According to numerical experiments, we set the value of µ equals to.5 for ships. As for the four models, the numbers of iterations are different mainly due to the different interior free surface. Through computing the ratio of area between the interior surface and wetted surface, we find that the ratios of cylinder, TLP, 2LNGs, S-175 are.14,.5,.65 and.58 respectively. According to the shapes of ships and platforms, the value of parameter µ is set to.2 when the ratio of area is less than.35. And the parameter µ should be set to.5 when the ratio of area is larger than.35. According to the above-mentioned results, it seems that the iterative method is always better than direct methods for the given linear system. But for some linear systems which have many right hand sides, it is important for us to select the more time saving solver. A flop is a floating point operation, i.e., a floating point addition or a floating point multiplication. As for a general linear system Ax = b, the dimension of A, x and b are N N, N M and N M, respectively. The flops of the direct method using classical LU factorization can be counted as follows: 2N 3 /3+2MN 2 (8) As for the ILU-GMRES, we divide it into two parts, including the incomplete LU decomposition and the process of iterations. For the first part, we set the parameter coef for calculation. The value of coef is about 1 2. The flops of the first part can be counted as coef 2N 3 /3. For the second part, applying M it for the number of iteration we can obtain: M [2N 2 +M it 4N 2 +M it (M it +1) 2N] (9) According to the numerical experiments, the M it is usually much less than N and larger than 1. The total flops of ILU-GMRES can be counted as follows. Let (8) equals to (1), we can obtain: coef 2N 3 /3+4N 2 M M it (1) M/N = (1 coef)/(6m it 3) (11) According to our numerical experiments, the value of coef is much less than 1 so that c = (6M it 3))M/N 1 (12) The above equation means the direct method and ILU-GMRES will cost the same CPU time. For larger N and less iterations M it, the criterion c < 1 and the ILU-GMRES method is preferable. On the other side, if more second members (larger M) and smaller N, the criterion c > 1 which means the direct method is more economic. According to the value of c, we will select ILU-GMRES for c < 1 and the direct method for c > 1. In order to validate the above criterion, we use the case of cylinder as a test. We select N = 3372, M = 66, and wave frequency is 2.3. After computing, we know M it = 9. Based on (12), we get c = 1.2 which means the iterative method consumes the same CPU time as with direct method. Actually, through our numerical experiment, the CPU time of direct method with LU factorization is 255 seconds and that of ILU-GMRES is 258 seconds. The number of iteration can not be known before the process of iteration. It is necessary for us to build a database about the number of iterations in different frequencies and shape of offshore structures. In the field of naval architecture and ocean engineering, the number of the directions of incident wave is usually less than 24 if we select an incident wave every 15 degree. That means the number of right hand sides is up to 3. According to our numerical experiments, the number of iteration is usually less than 15. Through the criterion, we can conclude that ILU-GMRES is always faster than direct method when the number of panels is larger than 26. On the other side, the number of the wave headings is usually larger than 7 and iteration is usually larger than 7 too. We conclude that direct methods are always faster than ILU-GMRES when the number of the panels is less than Conclusions For the given numerical model, ILU-GMRES is significantly faster than direct methods when solving the linear systems arising from boundary element analysis of offshore structures. By the preconditioner

6 with incomplete LU factorization, the number of iterations at high wave frequencies has been reduced effectively. The value of the parameter µ should be different according to different shape of offshore structures. When the ratio of area between interior free surface and wetted surface is less than.35, like semisubmersibles, TLP and spar, the value of µ should be equal to.2. When the value of ratio is larger than.35, the value of µ should be equal to.5. According to different linear systems, we should choose a better solver between ILU-GMRES method and direct method. When the number of the panels is larger than 26, we select ILU-GMRES. When the number of the panels is less than 27, we select direct methods. Between them, the criterion formula (12) can be useful to estimate which method is faster. Acknowledgments The research is financially supported by the National Natural Science Foundation of China (Grant No ), Excellent Youth Foundation of Heilongjiang Province of China and the 111 project (B719). References [1] Saad Y. and Schultz M.H. (1986) GMRES: A generalized minimal residual algorithm for solving non-symmetric linear systems. SIAM J. Sci. Stat. Comput. 7, [2] Kane J.H., Keyes D.E. and Prasad K.G (1991) Iteration solution techniques in boundary element analysis. Int. J. Numerical Methods Engng. 31, [3] Habashi W.G., Peeters M.F., Robichand M.P. and Nguyen V.N. (1992) A fully-coupled finite element algorithm using direct and iterative solvers for the incompressible Navier-Stokes equations. Incompressible computational fluid dynamics: trends and advances. Cambridge Univ. Press. ed. Gunzburger, M.D. and Nicolaides, R.A [4] Zhao Y. and Graham J.M.R. (1996) An iterative method for boundary solution of large offshore structures using the GMRES solver. Ocean Engng 23, [5] Makihata N., Utsunomiya T. and Watanabe E. (25) Effectiveness of GMRES-DR and OSP- ILUC for wave diffraction analysis of a very large floating structure (VLFS). Engng analysis with boundary elements 3, [6] Chen X.B. (24) Hydrodynamics in offshore and naval applications. Research Department, Bureau Veritas [7] Dai Y.S. and Duan W.Y. (28) Potential flow theory of ship motions in waves. National Defense Industry Press. [8] HydroStar for Experts - user manual (28) Research Department, Bureau Veritas. [9] Golub G.H. and Van Loan C.F. (1996) Matrix computations John Hopkins University Press. [1] Quarteroni A., Sacco R. and Saleri F. (2) Numerical mathematics, Springer [11] Eatock-taylor R. and Jeffery E.R. (1985) Variability of hydrodynamic load predictions for a tension leg platform. Ocean Engng. 13, [12] Dai Y.Z. (23) Rapid 3-dimensional hydroelastic analysis of large offshore structures. PhD thesis. College of Shipbuilding Engineering, Harbin Engineering University. [13] LAPACK routine DBDSQR with url as