Abstract
Fredholm integral equations of the second kind that arise in wave analysis of floating bodies are solved using a wavelet method. The two-dimensional linear wave-body problem for arrays of rectangular cylinders floating in the free surface of an otherwise unbounded fluid is considered. Both spline wavelets and the Daubechies wavelets with adaption to an interval are used as basis functions. An a priori compression strategy taking into account the singularities of the kernel of the integral equation, which arise at the corners of the geometry, is developed. The algorithm is O(n), where n is the number of unknowns. Computations of the hydrodynamic properties of the cylinders using the compression strategy are performed. The strategy is found to work well. A very high compression rate is obtained, still keeping a high accuracy of the computations. The accuracy of the potential close to the corners (singular points) is examined in a special case where an analytical solution is available.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. Büchmann, J. Skourup, and H. F. Cheung, Run-up on a structure due to second-order waves and a current in a numerical wave tank. Appl. Ocean Res. 20 (1998) 297-308.
S. Finne and J. Grue, On the complete radiation-diffraction problem and wave-drift damping of marine bodies in the yaw mode of motion. J. Fluid Mech. 357 (1998) 289-320.
J. N. Newman and C. H. Lee, Sensitivity of wave loads to the discretization of bodies. In: M. H. Patel and R. Gibbins (Eds), Proc. 6th Int. Conf. Behaviour of Offshore Structures (BOSS' 92). London: BPP Technical Services Ltd. (1992) 50-64.
C. H. Lee, H. D. Maniar, J. N. Newman, and X. Zhu, Computations of wave loads using a b-spline panel method. Twenty-first Symp. Naval Hydrodyn. (1996) 75-92.
H. D. Maniar and J. N. Newman, Wave diffraction by a long array of cylinders. J. Fluid Mech. 339 (1997) 309-330.
K. Mørken and J. O. Nygaard, Biorthogonal spline wavelets on an interval. Preprint. Dept. of Informatics, Univ. of Oslo. (2000) 35 pp.
G. Beylkin, R. Coifman, and V. Rokhlin, Fast wavelet transforms and numerical algorithms i. Communications on Pure and Applied Mathematics XLIV (1991) 141-183.
J. W. Wehausen and E. V. Laitone, Surface waves. In: S. Flugge, editor, Fluid Dynamics III, Encyclopedia of Physics. Berlin, Göttingen, Heidelberg: Springer-Verlag (1960) 446-778.
I. Daubechies, Ten Lectures on Wavelets. CBMS-NFS Regional conference series in applied mathematics. Philadelphia, Pennsylvania: SIAM (1992) 352 pp.
B. Jawerth and W. Sweldens, An overview of wavelet based multiresolution analyses. SIAM Rev. 36 (1994) 377-412.
A. Cohen, I. Daubechies, and P. Vial, Wavelets on the interval and fast wavelet transforms. Appl. Comp. Harmonic Anal. 1 (1993) 54-81.
W. Sweldens, The lifting scheme: A construction of second generation wavelets. Preprint, retrieved by ftp (http://cm.bell-labs.com/who/wim/papers/) (1995).
W. Sweldens and P. Schröder, Building your own wavelets at home. Retrieved by ftp (http://cm.bell-labs.com/who/wim/papers/) (1995).
A. M. Bruaset, A survey of preconditioned iterative methods. Pitman Research Notes inMathematics. Essex, UK: Longman Scientific & Technical (1995) 159 pp.
J. N. Newman and P. D. Sclavounos, Large-scale computation of wave loads on offshore structures. In: Faltinsen et al. (Ed.), Proc. 5th Int. Conf. Behaviour of Offshore Structures (BOSS' 88). Trondheim, Norway: Tapir (1988) 1-15.
H. Kober, Dictionary of Conformal Representations. USA: Dover (1952) 208 pp.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nygaard, J.O., Grue, J. Wavelet methods for the solution of wave-body problems. Journal of Engineering Mathematics 38, 323–354 (2000). https://doi.org/10.1023/A:1004750524689
Issue Date:
DOI: https://doi.org/10.1023/A:1004750524689