zbMATH — the first resource for mathematics

Boundary element formulation of the Mild-Slope equation for harmonic water waves propagating over unidirectional variable bathymetries. (English) Zbl 1403.76063
Summary: This paper presents a boundary element formulation for the solution of the Mild-Slope equation in wave propagation problems with variable water depth in one direction. Based on Greens function approximation proposed by K. A. Belibassakis [Wave Motion 32, No. 4, 339–361 (2000; Zbl 1074.76598)], a complete fundamental-solution kernel is developed and combined with a boundary element scheme for the solution of water wave propagation problems in closed and open domains where the bathymetry changes arbitrarily and smoothly in a preferential direction. The ability of the proposed formulation to accurately represent wave phenomena like refraction, reflection, diffraction and shoaling, is demonstrated with the solution of some example problems, in which arbitrary geometries and variable seabed profiles with slopes up to 1:3 are considered. The obtained results are also compared with theoretical solutions, showing an excellent agreement that demonstrates its potential.

76M15 Boundary element methods applied to problems in fluid mechanics
65N38 Boundary element methods for boundary value problems involving PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Full Text: DOI
[1] Belibassakis, K., The green׳s function of the mild-slope equationthe case of a monotonic bed profile, Wave Motion, 32, 339-361, (2000) · Zbl 1074.76598
[2] Berkhoff W. Computation of combined refraction - diffraction, In: Proceedings of 13th International Conference on Coastal Engineering, ASCE; 1972.
[3] Tsay, T. K.; Liu, P. L.-F., A finite element model for wave refraction and diffraction, Appl Ocean Res, 5, 1, 30-37, (1983)
[4] Booij, N., A note on the accuracy of the mild-slope equation, Coast Eng, 7, 3, 191-203, (1983)
[5] Kirby, J. T., A general wave equation for waves over rippled beds, J Fluid Mech, 162, 171-186, (1986) · Zbl 0596.76017
[6] Massel, S. R., Extended refraction-diffraction equation for surface waves, Coast Eng, 19, 97-126, (1993)
[7] Maa, J. P.Y.; Hsu, T. W.; Lee, D. Y., The RIDE modelan enhanced computer program for wave transformation, Ocean Eng, 29, 1441-1458, (2002)
[8] Chamberlain, P.; Porter, D., The modified mild-slope equation, J Fluid Mech, 291, 393-407, (1995) · Zbl 0843.76006
[9] Porter, D.; Staziker, D., Extensions of the mild-slope equation, J Fluid Mech, 300, 367-382, (1995) · Zbl 0848.76010
[10] Suh, K. D.; Lee, C.; Park, W. S., Time-dependent equations for wave propagation on rapidly varying topography, Coast Eng, 32, 91-117, (1997)
[11] Chandrasekera, C. N.; Cheung, K. F., Extended linear refraction-diffraction model, J Waterway Port Coast Ocean Eng, 123, 5, 280-286, (1997)
[12] Lee, C.; Park, W. S.; Cho, Y.-S.; Suh, K. D., Hyperbolic mild-slope equations extended to account for rapidly varying topography, Coast Eng, 34, 243-257, (1998)
[13] Copeland, G. J., A practical alternative to the “mild-slope” wave equation, Coast Eng, 9, 2, 125-149, (1985)
[14] Hsu, T. W.; Wen, C.-C., A parabolic equation extended to account for rapidly varying topography, Ocean Eng, 28, 11, 1479-1498, (2001)
[15] Li, B., An evolution equation for water waves, Coast Eng, 23, 3-4, 227-242, (1994)
[16] Hsu, T. W.; Wen, C. C., A study of using parabolic model to describe wave breaking and wide-angle wave incidence, J Chin Inst Eng, 23, 4, 515-527, (2000)
[17] Berkhoff W. Mathematical models for simple harmonic linear water waves. Wave diffraction and refraction [Ph.D. thesis]. Delft Hydraulics Laboratory; 1976.
[18] Li, B.; Anastasiou, K., Efficient elliptic solvers for the mild-slope equation using the multigrid technique, Coast Eng, 16, 3, 245-266, (1992)
[19] Panchang, V. G.; Pearce, B. R.; Wei, G.; Cushman-Roisin, B., Solution of the mild-slope wave problem by iteration, Appl Ocean Res, 13, 4, 187-199, (1991)
[20] Chen H, Mei C. Oscillations and wave forces in a man-made harbor in the open sea. In: Symposium on naval hydrodynamics 10th, Cambridge; 1974.
[21] Chen, H., Effects of bottom friction and boundary absorption on water wave scattering, Appl Ocean Res, 8, 2, 99-104, (1986)
[22] Tsay, T. K.; Zhu, W.; Liu, P.-F., A finite element model for wave refraction, diffraction, reflection and dissipation, Appl Ocean Res, 11, 1, 33-38, (1989)
[23] Bettess, P.; Zienkiewicz, O. C., Diffraction and refraction of surface waves using finite and infinite elements, Int J Numer Methods Eng, 11, 8, 1271-1290, (1977) · Zbl 0367.76014
[24] Lau, S.; Ji, Z., Efficient 3-D infinite element for water wave diffraction problems, Int J Numer Methods Eng, 28, 6, 1371-1387, (1989) · Zbl 0687.76009
[25] Givoli, D.; Keller, J. B., Non-reflecting boundary conditions for elastic waves, Wave Motion, 12, 3, 261-279, (1990) · Zbl 0708.73012
[26] Keller, J. B.; Givoli, D., Exact non-reflecting boundary conditions, J Comput Phys, 82, 1, 172-192, (1989) · Zbl 0671.65094
[27] Givoli, D., Non-reflecting boundary conditions, J Comput Phys, 94, 1, 1-29, (1991) · Zbl 0731.65109
[28] Bonet, R. P., Refraction and diffraction of water waves using finite elements with a DNL boundary condition, Ocean Eng, 63, 77-89, (2013)
[29] Beltrami, G.; Bellotti, G.; Girolamo, P.; Sammarco, P., Treatment of wave breaking and total absorption in a mild-slope equation FEM model, J Waterway Port Coast Ocean Eng, 127, 5, 263-271, (2001)
[30] Steward, D.; Panchang, V., Improved coastal boundary condition for surface water waves, Ocean Eng, 28, 1, 139-157, (2001)
[31] Chen W. Finite element modeling of wave transformation in harbors and coastal regions with complex bathymetry and ambient currents [Ph.D. thesis]. Department of Civil Engineering, University of Maine; August 2002.
[32] Liu, S. X.; Sun, B.; Sun, Z.-B.; Li, J.-X., Self-adaptive FEM numerical modeling of the mild-slope equation, Appl Math Model, 32, 12, 2775-2791, (2008) · Zbl 1167.86300
[33] Hauguel A. A combined FE-BIE method for water waves. In: 16th International conference on coastal engineering; 1978. p. 715-21.
[34] Shaw, R.; Falby, W., FEBIE—a combined finite element-boundary integral equation method, Comput Fluids, 6, 3, 153-160, (1978) · Zbl 0385.76027
[35] Hamanaka, K. I., Open, partial reflection and incident-absorbing boundary conditions in wave analysis with a boundary integral method, Coast Eng, 30, 281-298, (1997)
[36] Isaacson, M.; Qu, S., Waves in a harbour with partially reflecting boundaries, Coast Eng, 14, 3, 193-214, (1990)
[37] Lee, H. S.; Williams, A., Boundary element modeling of multidirectional random waves in a harbor with partially reflecting boundaries, Ocean Eng, 29, 1, 39-58, (2002)
[38] Lee, H. S.; Kim, S. D.; Wang, K.-H.; Eom, S., Boundary element modeling of multidirectional random waves in a harbor with a rectangular navigation channel, Ocean Eng, 36, 17-18, 1287-1294, (2009)
[39] Zhu, S., A new DRBEM model for wave refraction and diffraction, Eng Anal Bound Elem, 12, 4, 261-274, (1993)
[40] Liu H. A modified GDRBEM model for wave scattering. In: International conference on estuaries and coasts; 2003. p. 749-55.
[41] Zhu, S. P.; Liu, H.-W.; Chen, K., A general DRBEM model for wave refraction and diffraction, Eng Anal Bound Elem, 24, 5, 377-390, (2000) · Zbl 0980.76056
[42] Zhu, S. P.; Liu, H.-W.; Marchant, T. R., A perturbation DRBEM model for weakly nonlinear wave run-ups around islands, Eng Anal Bound Elem, 33, 1, 63-76, (2009) · Zbl 1188.76233
[43] Hsiao, S. S.; Chang, C.-M.; Wen, C.-C., Solution for wave propagation through a circular cylinder mounted on different topography ripple-bed profile shoals using DRBEM, Eng Anal Bound Elem, 33, 11, 1246-1257, (2009) · Zbl 1244.76045
[44] Naserizadeh, R.; Bingham, H. B.; Noorzad, A., A coupled boundary element-finite difference solution of the elliptic modified mild slope equation, Eng Anal Bound Elem, 35, 1, 25-33, (2011) · Zbl 1259.76031
[45] Bergmann, P. G., The equation in a medium with a variable index of refraction, J Acoust Soc Am, 17, 4, 329-333, (1946)
[46] Belibassakis, K.; Athanassoulis, G. A., Three-dimensional greens function for harmonic water waves over a bottom topography with different depths at infinity, J Fluid Mech, 510, 267-302, (2004) · Zbl 1123.76010
[47] Abramowitz, M.; Stegun, I., Handbook of mathematical functions, (1965), Dover Publications, NY
[48] Wu, T., Boundary element acoustics, fundamentals and computer codes, 5th ed., vol. 3, (2000), WIT Press Boston
[49] Atalla, N.; Sgard, F., Finite element and boundary methods in acoustics and vibration, (2015), Taylor & Francis Group, CRC Press, Taylor & Francis Group, 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742
[50] Dominguez, J., Boundary elements in dynamics, (1993), CMP, Elsevier, CMP Southampton, co-published with, Elsevier Essex · Zbl 0790.73003
[51] Panchang, V.; Chen, B. X.W.; Schelenker, K.; Demirbilek, Z.; Okihiro, M., Exterior bathymetry effects in elliptic harbor wave models, J Waterway Port Coast Ocean Eng, 126, 71-78, (2000)
[52] McCamy R, Fuchs R. Wave forces on piles; a diffraction theory. Beach Erosion Board, Technical Memorandum, no. 69; 1954.
[53] Mei, C., The applied dynamics of ocean surface waves, (1983), John Wiley Chichester · Zbl 0562.76019
[54] Rodríguez-Tembleque, L.; González, J. A.; Cerrato, A., Partitioned solution strategies for coupled BEM-FEM acoustic fluid-structure interaction problems, Comput Struct, 152, 45-58, (2015)
[55] Berkhoff, J.; Booy, N.; Radder, A., Verification of numerical wave propagation models for simple harmonic linear water waves, Coast Eng, 6, 3, 255-279, (1982)
[56] Belibassakis, K.; Athanassoulis, G.; Gerostathis, T., Acoupled-mode model for the refraction-diffraction of linear waves over steep three-dimensional bathymetry, Appl Ocean Res, 23, 6, 319-336, (2001)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.