Yanlin Shao 1 Odd M. Faltinsen 1 Ship Hydrodynamics & Stability, Det Norsk Veritas, Norway Centre for Ships and Ocean Structures (CeSOS), NTNU, Norway 1
The state-of-the-art potential flow analysis: Boundary Element Method (BEM) A Cruise Ship A numerical model for BEM Unknowns distributed on wetted ship surface and water surface Tune the unknowns so that all boundary conditions and governing equations are satisfied Still time consuming when it comes to fully-nonlinear problems
Introduction: Boundary Element Method Using Boundary Element Method (BEM) is a strong tradition in Marine Hydrodynamics. Early examples: Bai & Yeung (1974) Many applications of BEM wave-body interactions ---- Linear; ----Weakly-nonlinear (e.g. nd order); Advantage: Discretize only on boundary surfaces Disadvantage: Dense matrix, CPU time and memory increases as N Applications of BEM in fully-nonlinear wave-body interactions are quite limited. The CPU time is an important consideration
Introduction: Volume Methods Volume methods are found to be very efficient in terms of CPU time. Advantages to operate with sparse matrix. Examples: FEM: Wu & Eatock Taylor (1995), Wang & Wu (001) FDM: Bingham & Zhang (007), Engsig-Karup et al. (009) Advantage: Operate with sparse matrix Frame 001 9 Jun 01 3d contour Disadvantage: Mesh generation in whole fluid domain Y Z X
Harmonic Polynomials Leonhard Euler (1707-1783) Euler s memoir Principia motus fluidorum (Euler, 1756 1757) English translation available at www.oca.eu/etc7/ee50/texts/euler1761eng.pdf
x, y, z 0 Use polynomials to represent velocity potential Constraints on coefficients in order to satisfy Laplace equation
Examples of Harmonic Polynomials In 1D: f x a bx n In D: z x i y n u x, y i v x, y Examples up to 4 th order: 1, x, y, x -y, xy, x 3-3xy, 3x y-y 3, x 4-6x y +y 4, 4x 3 y-4xy 3 n n Laplace equation automatically satisfied.
An example of using harmonic polynomials (D): A Dirichlet boundary-value problem 8 x y b f j x, y, j j1 Unknown coefficients Harmonic polynomials Applying boundary conditions at the nodes gives i b 8 j1 b 8 j i i, j j j1 f x x, y y, i 1,,8 c j i i, i 1,,8 6 7 8 0 4 5 1 3 i 8 8 d ( x, y) j, i j Linear combination of, 1,,8. i i i i1 j1 x, y c f x, y
The HPC method for a general potential-flow problem (i-1,j+1) (i-1,j) (i,j+1) (i+1,j+1) (i+,j+1) (i,j) (i+1,j) (i+,j) (i-1,j-1) (i,j-1) (i+1,j-1) (i+,j-1) 1. Discretize by quadrilateral elements. Operate with cells that contain 4 neighboring quadrilateral elements and 9 grid points 3. Consider a sub-dirichlet problem in each cell In fluid: 8 8 j, i j i i1 j1 x, y c f x, y 6 7 8 9 4 5 9 x x 0, y y 0 c ii 9 9 1, i1 On Neumann boundaries: 8 8 x, y c j, if j x, yn( x, y) i n i1 j1 8 1 3 Sparse matrix with at most 9 nonzeros in each row. 3 rd ~4 th order accuracy.
Efficiency & Accuracy
A D analytical case Dirichlet surface Neumann 0 L h Neumann Neumann Length = L, Height = h, L = 40h (Consistent with Wu & Eatock Taylor, 1995) Uniform rectangular grids x y Analytical velocity potential cosh k( y h) sin kx Mixed Dirichlet-Neumann boundary value problem GMRES solver used for all the methods in comparisons
L errors L Errors CPU time (s) CPU time 10 BEM FMM-BEM FVM LPC HPC N: number of unknowns corresponding to BEMs For a given accuracy, HPC performs best 1 Required CPU time to achieve 10-4 accuracy 0.1 10 L 1 10 0 errors 10-1 10 3 x10 3 3x10 3 Number of unknowns FMM-BEM, Dirichlet surface FMM-BEM, Neumann surface HPC- D N 10 1 L 10 0 errors 10-1 HPC: 0.06 sec FMM-BEM: > 1 sec BEM: much much longer time 10-10 - 10-3 10-4 10-5 10-6 10-7 kh=1.0 FVM LPC HPC 10-3 10-4 10-5 10-6 10-7 FVM LPC HPC FMM-BEM, Dirichlet surface FMM-BEM, Neumann surface kh=6.8 10-8 10 3 x10 3 3x10 3 Number of unkowns N 10-8 10 3 x10 3 3x10 3 Number of Unknowns N
F ra m e 0 0 1 1 7 A pr 0 1 pa ne l on e pisode solid A 3D analytical case: a cube Z Dirichlet surface X Y Analytical velocity potential: sin k x k y exp( k z) x y z k 0.5, k 0.5, k k k x y z x y Uniform grid: h x y z Neumann surface
L errors CPU time (s) Comparison with Quadratic BEM (QBEM) and Fast Multipole Accelerated BEM (FMA-QBEM) L 10-3 errors 10-4 10-5 k =.71 10 4 CPU time 10 3 10 QBEM FMA-QBEM,p=1 FMA-QBEM,p=15 HPC k =.08 k = 1.4 k = 1.335 10-6 10-7 10-8 k = 3.610 k = 3.346 k = 3.896 S D, QBEM S N, QBEM S D, HPC S N, HPC 10 1 10 0 k = 1.78 10-9 10-10 -1 10 0 Element size h=x=y=z QBEM 3 L error h.71 7.968 10 D QBEM 3 L error h 3.610 4.038 10 N HPC L error h 3.346 1.371 10 D HPC 3 L error h 3.896 4.014 10 N 10-1 x10 3 4x10 3 6x10 3 8x10 3 10 4 Number of Unknowns S D = Dirichlet surface S N = Neumann surface N
Applications of HPC-D 15
wave maker Frame 001 16 Feb 01 contour lines Nonlinear numerical wave tank free surface bottom Piston wave maker The grid is updated to conform the deformation of the free surface
wave amplutude (m) Trial h (m) e (m) T (s) α β U r C 0.4 0.113 3.5 0.105 0.059 30. 0.06 0.05 0.04 0.03 0.0 0.01 1 st,num, fw=0.0 nd,num,fw=0.0 3 rd,num,fw=0.0 4 th,num,fw=0.0 1 st,exp nd,exp 3 rd,exp 4 th,exp 1 st,num,fw=0.05 nd,num,fw=0.05 3 rd,num,fw=0.05 4 th,num,fw=0.05 0.00 0 5 10 15 0 5 Distance to wave maker (m) A Rayleigh damping term is introduced in the dynamic free surface condition 4e 3 h f w fw = 0.05 is used 0% of that suggested by Chapalain et al. (199) Non-negligible damping effects for higher harmonics A more rational way of estimating damping effect is needed
(m) (m) (m) (m) Nonlinear waves over submerged trapezoidal bar Experiments results available from : Beji & Battjes (1993), Luth et al. (1994) 0.4 m 0.3 m 1:0 1.5 m 17.3 m 14.5 m 1 m 1:10 6 m 6 m m 3 m 13 m 0.04 0.03 Num. Exp. x = 1.5 m 0.04 0.03 Num. Exp. x = 14.5 m 0.0 0.0 0.01 0.00-0.01 0.01 0.00-0.01-0.0 0 1 3 4 5 6 t (s) -0.0 6 7 8 9 30 31 3 t (s) 0.04 0.03 Num. Exp. x = 17.3 m 0.04 0.03 Num. Exp. x = 1 m 0.0 0.0 0.01 0.01 0.00 0.00-0.01-0.01-0.0 8 9 30 31 3 33 34 t (s) -0.0 35 36 37 38 39 40 41 t (s)
Applications of HPC-3D 19
Fully-nonlinear wave tank (HPC-3D) Y HPC results agree surface well with experiments Water Sea floor 0 0 0 X
wave amplitude (m) wave amplitude (m) 0.04 1 st harmonic,exp. ra m e 0 0 1 0 9 M a1 st yharmonic,num. 0 1 3 d contour nd harmonic,exp. 0.00 nd harmonic,num. 3 rd harmonic,num. 3 rd harmonic,exp. 0.016 0.016 0.01 1 st harmonic,num. nd harmonic,num. 3 rd harmonic,num. 1 st harmonic,exp. nd harmonic,exp. 3 rd harmonic,exp. 0.01 0.008 0.008 0.004 0.004 0.000 0 5 10 15 0 5 30 35 x (m) 0.000 0 5 10 15 0 5 30 35 x (m) T = s; ka = 0.01 T = s; ka = 0.017
Fully-nonlinear wave diffraction (HPC-3D) F ra m e 0 0 1 J a n 0 1 3 4 - n o d e s F E M p a n e ls F E - V o lu m e B ric k D a ta Higher order horizontal wave forces in harmonic waves Comparisons with numerical (Ferrant) and experimental (Huseby&Grue) results
3 A third and even higher order wave load effect in survival conditions
Arg(F ) Arg(F 4 ) F /ga R Arg(F 1 ) Arg(F 3 ) F a F 1 /gar gar arg F 7. 7.0 6.8 6.6 6.4 6. 6.0 0.00 0.05 0.10 0.15 0.0 0.5 F a ga R arg F.4.0 1.6 1. 0.8 4 Analytical Experiment Analytical Experiment Present Ferrant Present Ferrant 0.00 0.05 0.10 0.15 0.0 0.5 0.7 0.6 0.5 0.4 0.3 0. 0.1 Analytical Experiment Present Ferrant 0.0 0.00 0.05 0.10 0.15 0.0 0.5 3.0.5.0 1.5 1.0 0.5 kr k R A 0.45, /g=wave number, radius, wave amplitude Analytical Experiment Present Ferrant 0.0 0.00 0.05 0.10 0.15 0.0 0.5 A ka ka ka ka 3 arg F F a F 3 /ga 3 3 ga 4 3 F a 4 1 ga R F 4 /ga 4 R -1 4 arg F 0.4 0.3 0. 0.1 0.0 0.00 0.05 0.10 0.15 0.0 0.5 4 3 0.5 0.4 0.3 0. 0.1 3 1 0-1 - Analytical Experiment Analytical Experiment -3 0.00 0.05 0.10 0.15 0.0 0.5 A Present Ferrant Experiment Ferrant Present 1 0.00 0.05 0.10 0.15 0.0 0.5 A A Present Ferrant 0.0 0.00 0.05 0.10 0.15 0.0 ka 0.5 ka ka ka Experiment Ferrant Present
Current effect on wave run-up.4.4.0 Ferrant 001, Fr = -0.05 Present.0 Ferrant 001, Fr = 0.05 Present max H 1.6 1. max H 1.6 1. 0.8 Ferrant, Fr =0.05 0.8 0.4 Present 0.4 Ferrant 001, Fr = -0.05 Present 0.0 0.0 0. 0.4 0.6 0.8 1.0 0.0 0.0 0. 0.4 0.6 0.8 1.0 Fr = 0.05 Fr = -0.05 Comparison for wave runup around cylinder 5 5
elevation at P 1 (m) elevation at P 1 (m) Sloshing 0.08 0.04 0.90 1 0.00-0.04 HPC Exp. -0.08 0 4 8 1 16 0 4 8 time (s) 0.10 0.05 0.93 1 0.00-0.05 HPC Exp. -0.10 0 10 0 30 40 time (s) Forced oscillation: X T = 0.005L cos(30 o ), Y T = 0.005L sin(30 o )
Limitations of the HPC method Vortex shedding e.g. Bilge keel Wave breaking 7
A clever strategy: Domain Decomposition Potential-flow solver Efficiency Accuracy More advanced solver Breaking Fragmentation Air entrainment Viscosity 8
Thank you!
References Bai K. J., Yeung R.W., Numerical solutions to free-surface flow problems, Proceedings of 10 th Symposium on Naval Hydrodynamics, Cambridge, MA, 1974. Wu GX, Eatock Taylor (1995) Time stepping solutions of the two-dimensional nonlinear wave radiation problem. Ocean Engng. (8), 785-798. Bingham H.B., Zhang H., On the accuracy of finite difference solutions for nonlinear water waves. J. Engineering Math., 58, 11-8, 007. Chapalain G, Cointe R, Temperville A., Observed and modeled resonantly interacting progressive water-waves. Costal Engineering, 16, 67-300, 199. Luth H.R., Klopman G., Kitou N., Kinematics of waves breaking partially on an offshore bar; LDV measurements of waves with and without a net onshore current. Report H-1573, Delft Hydraulics, 1994. P. Ferrant, Fully nonlinear interactions of long-crested wave packets with a three dimensional body. In Proceedings of nd ONR Symp. in Naval Hydrodynamics, 59-7, 1998. A.P. Engsig-Karup, H.B. Bingham, O. Lindberg, An efficient flexible-order model for 3D nonlinear water waves. J. Comput. Phys. 8 (009) 100-118. C.H.Wu, O.M. Faltinsen, B.F. Chen, Time-independent finite difference and ghost cell method to study sloshing liquid in d and 3d tanks with internal structures. Commun. Comput. Phys. 13 (013) 780-800. Ferrant P., Runup on a cylinder due to waves and current: potential flow solution with fully nonlinear boundary conditions. International Journal of Offshore and Polar Engineering. 11(1), 001. Euler L., Principles of the motion of fluids, Physica D: Nonlinear Phenomena 37 (008) 1840-1854. Shao Y.L., Faltinsen O.M., Towards efficient fully-nonlinear potential-flow solvers in marine hydrodynamics. in: Proc. of the 31st Int. Conf. on Ocean, Offshore and Arc. Eng. (OMAE). Rio de Janeiro, Brazil, 01. Shao Y.L., Faltinsen O.M., A Harmonic polynomial cell (HPC) method for 3D Laplace equation with application in marine hydrodynamics. Submitted for journal publication. Shao Y.L., Faltinsen O.M., Use of body-fixed coordinate system in analysis of weakly-nonlinear wave-body problems. Appl. Ocean Res., 3, 1, 0-33, 010.